Answer:
3.573 to 4.127
Step-by-step explanation:
Given
Sample size = 15
Mean = Sum of ratings/ sample size
Mean = 57.7/15
Mean = 3.85
Degree of freedom = sample size - 1
Degree of freedom = 15 - 1 = 14
df = 14
Then we calculate the standard deviation
(x - mean)² ||
(3.6 - 3.85)² || 0.0625
(2.9 - 3.85)² || 0.9025
(3.8 - 3.85)² || 0.0025
(4.5 - 3.85)² || 0.4225
(3.2 - 3.85)² || 0.4225
( 3.9 - 3.85)² || 0.0025
( 3.3 - 3.85)² || 0.3025
( 4.6 - 3.85)² || 0.5625
(4.1 - 3.85)² || 0.0625
(4.3 - 3.85)² || 0.2025
4.4 - 3.85)² || 0.3025
( 3.9 - 3.85)² || 0.0025
(3.2 - 3.85)² || 0.4225
( 4.2 - 3.85)² || 0.1225
( 3.8 - 3.85)² || 0.0025
Total || 3.7975
Variance = 3.7975/15 = 0.253167
Standard Deviation = √0.253167 = 0.50315703314174194
Standard Deviation = 0.5 ------- Approximated
The next step is to subtract the confidence level from 1, then divide by two.
i.e (1 - 0.95)/2 = 0.025
α = 0.025
Then we look up this answer to step in the t-distribution table.
For 14 degrees of freedom (df) and α = 0.025, my result is 2.145
The next step is to divide the sample standard deviation by the square root of the sample size.
0.5 / √15 = 0.129
Next is to multiply this result by step 2.145 (from the t table)
0.129 × 2.45 = 0.277
For the lower end of the range, subtract 0.277 from the sample mean.
3.85 – 0.277 = 3.573
Step 7: For the upper end of the range, add step 0.277 to the sample mean.
3.85 + 0.277 = 4.127
A customer at a self -storage facility was offered a choice between a storage unit shaped like a cube and another unit is 2 feet longer,5 feet shorter than the first unit. The customer thinks that f the volume of the cube is x^3 the volume of the other unit would be x^3-4x^2-11x+30. Is the customer correct ?
Answer: No, the Volume is x^3 - 3x^2 - 10x
Step-by-step explanation:
Since the volume of the cubic storage unit is x^3
Therefore,
Length = x
Width = x
Height = x
For the new storage unit
Length = x + 2
Width = x
Height = x - 5
Volume = ( x + 2)(x)(x -5)
V = x (x^2 -3x - 10)
V = x^3 - 3x^2 - 10x
Therefore, the volume of the new storage unit is x^3 - 3x^2 - 10x
Answer:the customer is incorrect
Step-by-step explanation:
In a cube, all 4 sides are equal. The volume of a cube that has x as the length of each side would be x^3
If the customer thinks that f the volume of the cube is x^3, it means that each side is x. Then the other storage unit offered to the customer is 2 feet longer,5 feet shorter than the first unit. Its dimensions would be (x+ 2) feet, (x - 5) feet and x feet
The volume of the other storage unit should be
x[(x + 2)(x - 5)] = x(x^2 - 5x + 2x + 10)
= x(x^2 - 3x + 10)
= x^3 - 3x^2 + 10x
Which of the following statements are true of hypothesis tests?
1.You must state null and alternative hypotheses in the context of the problem.
2.You must state a significance level so you can decide if a given P-value gives you evidence to reject the null hypothesis.
3.You must state a conclusion in the context of the problem.
In hypothesis testing, it is critical to state the null and alternative hypotheses, choose an appropriate significance level, and conclude in the context of the problem. Decisions must reflect the probabilistic nature of the tests, with careful consideration of Type I and Type II errors.
In hypothesis testing, the following statements are indeed true:
You must state null and alternative hypotheses in the context of the problem.
You must state a significance level so you can decide if a given P-value provides evidence to reject the null hypothesis.
You must state a conclusion in the context of the problem.
When conducting a hypothesis test, one must also be mindful not to claim that a hypothesis is definitively proven true or false due to the probabilistic nature of hypothesis testing. Instead, you can infer whether there is sufficient evidence to support the alternative hypothesis if the null hypothesis is rejected. However, remember that making a decision at a certain significance level involves a trade-off between Type I and Type II errors.
a. Write an equation that represents the sum of the angle measures of the triangle.
b. Use your equation and the equation shown to find the values of x and y.
The Triangle Angle Sum Theorem states that the sum of interior angles in any triangle is always 180 degrees. Represented by the equation x + y + z = 180°, it allows for solving missing angles in a triangle using x = 180° - y - z or similar expressions.
Understanding the Triangle Angle Sum Theorem:
In any triangle, regardless of its shape or size, the sum of the interior angles always equals 180 degrees. This is known as the Triangle Angle Sum Theorem.
This theorem is a fundamental property of triangles and has numerous applications in geometry and other mathematical fields.
Representing the Angle Sum with an Equation:
Let's use variables to represent the angle measures of a triangle:
Angle 1 = x
Angle 2 = y
Angle 3 = z
According to the Triangle Angle Sum Theorem, the equation becomes:
x + y + z = 180°
Solving for Missing Angles:
This equation can be used to solve for any missing angle if we know the values of the other two angles.
For example, if we know the measures of angles y and z, we can find x using:
x = 180° - y - z
Similarly, we can find y or z if we know x and the other angle.
Example:
Consider a triangle with angles x = 50°, y = 70°, and z unknown.
Using the equation:
z = 180° - x - y = 180° - 50° - 70° = 60°
The equation that represents the sum of the angle measures of the triangle is 2y + x = 198.
The value of x is 86 and the value of y is 56.
A)
The sum of the interior angles of a triangle adds up to 180 degrees.
Hence, the equation that represents the sum of the angle measures of the given triangle is:
( y - 18 ) + y + x = 180
Simplifying; we get:
y + y + x = 180 + 18
2y + x = 198
B)
To solve for the values of x and y, we solve the system of equations:
2y + x = 198
3x - 5y = -22
Solve for x in equation 1:
2y + x = 198
x = -2y + 198
Plug x = -2y + 198 into equation 2 and solve for y:
3( -2y + 198 ) - 5y = -22
-6y + 594 - 5y = -22
-11y + 594 = -22
11y = 594 + 22
11y = 616
y = 616/11
y = 56
Now, plug y = 56 into equation 3 and solve for x:
x = -2y + 198
x = -2( 56 ) + 198
x = -112 + 198
x = 86
Therefore, the x = 86 and y = 56.
The missing image is uploaded below:
Nationwide, the average waiting time until a electric utility customer service representative answers a call is 200 seconds per call. The Gigantic Kilowatt Energy Company took a sample of 30 calls and found that, on the average, they answered in 120 seconds per call. Moreover, it is know that the standard deviation of the times for all such calls is 25 seconds. At the .05 significance level, is there evidence that this company's mean response time is lower than the average utility?
Answer:
[tex]z=\frac{120-200}{\frac{25}{\sqrt{30}}}=-17.527[/tex]
[tex]p_v =P(Z<-17.527) \approx 0[/tex]
If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis.
We can say that at 5% of significance the mean average waiting time is significantly less than 200 seconds per call.
Step-by-step explanation:
Data given and notation
[tex]\bar X=120[/tex] represent the sample mean
[tex]\sigma=25[/tex] represent the population standard deviation
[tex]n=30[/tex] sample size
[tex]\mu_o =200[/tex] represent the value that we want to test
[tex]\alpha=0.05[/tex] represent the significance level for the hypothesis test.
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value for the test (variable of interest)
State the null and alternative hypotheses.
We need to conduct a hypothesis in order to check if the population mean is less than 200, the system of hypothesis are :
Null hypothesis:[tex]\mu \geq 200[/tex]
Alternative hypothesis:[tex]\mu < 200[/tex]
Since we know the population deviation, is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:
[tex]z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}[/tex] (1)
z-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".
Calculate the statistic
We can replace in formula (1) the info given like this:
[tex]z=\frac{120-200}{\frac{25}{\sqrt{30}}}=-17.527[/tex]
P-value
Since is a one-side left tailed test the p value would given by:
[tex]p_v =P(Z<-17.527) \approx 0[/tex]
Conclusion
If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis.
We can say that at 5% of significance the mean average waiting time is significantly less than 200 seconds per call.
Approximately how much principal would need to be placed into an account earning 3.575% interest compounded quarterly so that it has an accumulated value of $68,000 at the end of 30 years
Answer: the principal is approximately 23377
Step-by-step explanation:
Let the Initial amount deposited into the account be $x This means that the principal is P = $x
It was compounded quarterly. This means that it was compounded four times in a year. So
n = 4
The rate at which the principal was compounded is 3.575%. So
r = 3.575/100 = 0.03575
It would be compounded for 30 years So
t = 30
The formula for compound interest is
A = P(1+r/n)^nt
A = total amount in the account at the end of t years.
A is given as $68,000
Therefore
68000= x (1+0.03575/4)^4×30
68000= x (1+0.0089375)^120
68000= x (1.0089375)^120
68000 = 2.90878547719x
x = 68000/2.90878547719
x = 23377.4545
Answer:
The answer is B.
Step-by-step explanation:
23,377.
a tree casts a shadow 8 feet long. A 6-foot Man cast a shadow 4 feet long. The triangle formed by the tree and its shadow is similar to the triangle formed by the man and his shadow. How tall is the tree?
Answer:
12 feet
Step-by-step explanation:
Draw a diagram (see picture below). The tree and its shadow is one triangle, the man and its shadow is another triangle. We assume both are right triangles because people and trees stand vertical.
Create a proportion to solve. Put the missing value in a numerator.
Tree height / Tree shadow = Man height / Man shadow
[tex]\frac{x}{8} =\frac{6}{4}[/tex]
Solve using cross multiplication. Multiply x by 4. Multiply 6 by 8.
4x = 48 Divide both sides by 4 to isolate x.
x = 12 Height of tree
The tree is 12 feet tall.
The number of customers that enter a store during one day in an example of :
-a continuous random variable
-a discrete random variable
-either a continuous or a discrete random variable, depending on the number of the customers
-either a continuous or a discrete random variable, depending on the gender of the customers
Answer:
a discrete random variable
Step-by-step explanation:
You can only have a natural number of clients entering the store.
For example, 0 clients, 1 client, 2 clients, 100 clients, ...
You cannot have a decimal value, for example, 0.5 clients.
So the correct answer is:
a discrete random variable
Much of what we know about left and right hemisphere specializations comes from the study of people who had split-brain surgery. This surgery _____ Select one:
a. splits the lobes of the brain apart.
b. severs the corpus callosum between hemispheres.
c. severs the nerves from the spinal cord to the right hemisphere.
d. severs the substantia nigra between hemispheres.
Answer:
b. severs the corpus callosum between hemispheres.
Step-by-step explanation:
The split-brain surgery is used to alleviate epileptic seizures. It involves the severing of the corpus callosum, that is the bond between both hemispheres of the brain.
So the correct answer is:
b. severs the corpus callosum between hemispheres.
A process is normally distributed with a mean of 104 rotations per minute and a standard deviation of 8.2 rotations per minute. If a randomly selected minute has 118 rotations per minute, would the process be considered in control or out of control?A. In control as only one data point would be outside the allowable rangeB. In control as this one data point is not more than three standard deviations from the meanC. Out of control as this one data point is more than three standard deviations from the meanD. Out of control as this one data point is more than two standard deviations from the mean
Answer:
Option B) In control as this one data point is not more than three standard deviations from the mean
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 104 rotations per minute
Standard Deviation, σ = 8.2 rotations per
We are given that the distribution of process is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
For x = 118
[tex]z = \displaystyle\frac{118-104}{8.2} = 1.7073[/tex]
Thus, we could say that this data point lies within three standard deviations from the mean as:
[tex]\mu - 3\sigma < x < \mu + 3\sigma\\104-3(8.2) < x < 104 + 3(8.2)\\79.4 < 118 < 128.6[/tex]
Thus, it could be said
Option B) In control as this one data point is not more than three standard deviations from the mean
The process would be considered out of control as the randomly selected minute has more than two standard deviations away from the mean.
Explanation:To determine whether the process is in control or out of control, we can use the Empirical Rule. The Empirical Rule states that approximately 68 percent of the data is within one standard deviation of the mean, approximately 95 percent of the data is within two standard deviations of the mean, and more than 99 percent of the data is within three standard deviations of the mean. In this case, since the randomly selected minute has 118 rotations per minute, which is more than two standard deviations away from the mean (104 rotations per minute), the process would be considered out of control.
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Find the average value of the function over the given solid. The average value of a continuous function f(x, y, z) over a solid region Q is 1 V Q f(x, y, z) dV where V is the volume of the solid region Q. f(x, y, z) = x + y + z over the tetrahedron in the first octant with vertices (0, 0, 0), (2, 0, 0), (0, 2, 0) and (0, 0, 2).
Compute the volume of [tex]Q[/tex]:
[tex]\displaystyle\iiint_Q\mathrm dV=\int_0^2\int_0^{2-x}\int_0^{2-x-y}\mathrm dz\,\mathrm dy\,\mathrm dx=\frac43[/tex]
Integrate [tex]f(x,y,z)=x+y+z[/tex] over [tex]Q[/tex]:
[tex]\displaystyle\iiint_Qf(x,y,z)\,\mathrm dV=\int_0^2\int_0^{2-x}\int_0^{2-x-y}(x+y+z)\,\mathrm dz\,\mathrm dy\,\mathrm dx=2[/tex]
So the average value of [tex]f[/tex] over [tex]Q[/tex] is 2/(4/3) = 3/2.
To solve this mathematical problem, we need to understand the Average Value of a Continuous function.
What is the Average Value of a Continuous Function?
The average value of a continuous function is derived by taking the integral of the function over the interval. This is then divided using the length of that interval.
How do we find the average value of the function?To determine the average value of the function f(x, y, z), over the solid region named Q,
we can say:
[tex]\int\int\int _{Q}[/tex] dV = [tex]\int_{0}^{2} \int_{0}^{2-x} \int_{0}^{2-x-y}[/tex] dzdydx = 4/3
Integrating the above, we have
[tex]\int\int\int _{Q}[/tex] [tex]f(x,y,z)[/tex] dV = [tex]\int_{0}^{2} \int_{0}^{2-x} \int_{0}^{2-x-y}[/tex] (x+ y + z) dzdydx = 2
Therefore, the average value of the function f over the Solid region Q becomes:
2/ (4/3) = 1.5 or 3/2
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For women aged 18-24, systolic blood pressures are normally distributed with a mean of 114.8 mm Hg and a standard deviation of 13.1 mm Hg. If 23 women aged 18-24 are randomly selected, find the probability that their mean systolic blood pressure is between 119 and 122 mm Hg. Round to four decimal places.
The probability is approximately 0.0833, indicating an 8.33% chance that the mean falls between 119 and 122 mm Hg.
To determine the probability that the mean systolic blood pressure of 23 randomly selected women aged 18-24 falls between 119 and 122 mm Hg, we utilize the Central Limit Theorem and z-scores.
First, we calculate the standard error of the mean (SEM) using the population standard deviation and the sample size. With a mean of 114.8 mm Hg and a standard deviation of 13.1 mm Hg, the SEM is approximately 2.7316 mm Hg.
Then, we standardize the values of 119 and 122 mm Hg into z-scores. For 119 mm Hg, the z-score is approximately 0.3206, and for 122 mm Hg, it's approximately 0.5496.
Using a standard normal distribution table or calculator, we find the area under the curve between these z-scores, representing the probability. Subtracting the cumulative probability of the lower z-score from the higher z-score gives us approximately 0.0833. This indicates that there's an 8.33% chance that the mean systolic blood pressure of the 23 randomly selected women aged 18-24 falls between 119 and 122 mm Hg. Thus, within the specified range, there's a moderate probability of occurrence based on the given parameters of the population distribution.
A survey finds that 55 people out of 170 favor increasing property taxes to help pay for a new library. If this data is used to estimate the population proportion who favor new taxes, the standard error of the estimate is:
A. 0.425
B. 0.036
C. 0.324
D. 0.001
E. 0.119
Answer: B. 0.036
Step-by-step explanation:
Formula for standard error :
[tex]SE=\sqrt{\dfrac{p(1-p)}{n}}[/tex]
, where p = Population proportion and n= sample size.
Let p be the population proportion of the people who favor new taxes.
As per given , we have
n= 170
[tex]p=\dfrac{55}{170}\approx0.324[/tex]
Substitute these values in the formula, we get
[tex]SE=\sqrt{\dfrac{0.324(1-0.324)}{170}}\\\\=\sqrt{0.00129}\\\\=0.0359165699921\approx0.036[/tex]
Hence, the standard error of the estimate is 0.036.
∴ The correct answer is OPTION B. 0.036
Solve the proportion. When necessary, round to the nearest tenth? 36/j = 7/20
Work is provided in the image attached.
A real estate agent would like to predict the selling price of a single-family house by predicting the price (in thousands of dollars) based on the square footage (in 100 square feet). If the LSRL for the data is ? = 3.8785x + 18.3538, predict the price of a 4000 square foot house (in thousands of dollars). tbl
Answer:
$173493.8
Step-by-step explanation:
Data provided in the question:
LSRL for the data is ? = 3.8785x + 18.3538
Here,
x is area in 100 square feet
and
price in thousands of dollar
Thus,
For the given area 4000 square foot
x = 4000 ÷ 100 = 40 [Area in 100 square feet]
Therefore,
Using the given equation
Price = 3.8785(40) + 18.3538
or
Price = 173.4938 in thousands of dollar
or
Price = 173.4938 × $1000
Price = $173493.8
Define the double factorial of n, denoted n!!, as follows:n!!={1⋅3⋅5⋅⋅⋅⋅(n−2)⋅n} if n is odd{2⋅4⋅6⋅⋅⋅⋅(n−2)⋅n} if n is evenand (−1)!! =0!! =1.Find the radius of convergence for the given power series.[(8^n*n!*(3n+3)!*(2n)!!)/(2^n*[(n+9)!]^3*(4n+3)!!)]*(8x+6)^n
Answer:
Radius of convergence of power series is [tex] \lim_{n \to \infty}\frac{a_{n}}{a_{n+1}}=\frac{1}{108}[/tex]
Step-by-step explanation:
Given that:
n!! = 1⋅3⋅5⋅⋅⋅⋅(n−2)⋅n n is odd
n!! = 2⋅4⋅6⋅⋅⋅⋅(n−2)⋅n n is even
(-1)!! = 0!! = 1
We have to find the radius of convergence of power series:
[tex]\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{2^{n}[(n+9)!]^{3}(4n+3)!!}](8x+6)^{n}\\\\\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{2^{n}[(n+9)!]^{3}(4n+3)!!}]2^{n}(4x+3)^{n}\\\\\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{[(n+9)!]^{3}(4n+3)!!}](x+\frac{3}{4})^{n}\\[/tex]
Power series centered at x = a is:
[tex]\sum_{n=1}^{\infty}c_{n}(x-a)^{n}[/tex]
[tex]\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{2^{n}[(n+9)!]^{3}(4n+3)!!}](8x+6)^{n}\\\\\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{2^{n}[(n+9)!]^{3}(4n+3)!!}]2^{n}(4x+3)^{n}\\\\\sum_{n=1}^{\infty}[\frac{8^{n}4^{n}n!(3n+3)!(2n)!!}{[(n+9)!]^{3}(4n+3)!!}](x+\frac{3}{4})^{n}\\[/tex]
[tex]a_{n}=[\frac{8^{n}4^{n}n!(3n+3)!(2n)!!}{[(n+9)!]^{3}(4n+3)!!}]\\\\a_{n+1}=[\frac{8^{n+1}4^{n+1}n!(3(n+1)+3)!(2(n+1))!!}{[(n+1+9)!]^{3}(4(n+1)+3)!!}]\\\\a_{n+1}=[\frac{8^{n+1}4^{n+1}(n+1)!(3n+6)!(2n+2)!!}{[(n+10)!]^{3}(4n+7)!!}][/tex]
Applying the ratio test:
[tex]\frac{a_{n}}{a_{n+1}}=\frac{[\frac{32^{n}n!(3n+3)!(2n)!!}{[(n+9)!]^{3}(4n+3)!!}]}{[\frac{32^{n+1}(n+1)!(3n+6)!(2n+2)!!}{[(n+10)!]^{3}(4n+7)!!}]}[/tex]
[tex]\frac{a_{n}}{a_{n+1}}=\frac{(n+10)^{3}(4n+7)(4n+5)}{32(n+1)(3n+4)(3n+5)(3n+6)+(2n+2)}[/tex]
Applying n → ∞
[tex]\lim_{n \to \infty}\frac{a_{n}}{a_{n+1}}= \lim_{n \to \infty}\frac{(n+10)^{3}(4n+7)(4n+5)}{32(n+1)(3n+4)(3n+5)(3n+6)+(2n+2)}[/tex]
The numerator as well denominator of [tex]\frac{a_{n}}{a_{n+1}}[/tex] are polynomials of fifth degree with leading coefficients:
[tex](1^{3})(4)(4)=16\\(32)(1)(3)(3)(3)(2)=1728\\ \lim_{n \to \infty}\frac{a_{n}}{a_{n+1}}=\frac{16}{1728}=\frac{1}{108}[/tex]
1) The sum of a sequence of consecutive integers is 342. The largest integer in the sequence is 3 times greater than the smallest integer in the sequence. What is the smallest integer and how many integers are in the sequence?
Due tmr tysm
Answer:
The smallest integer is 9 and there are 19 terms in the sequence.
Step-by-step explanation:
Arithmetic Sequence
The general term of an arithmetic sequence is
[tex]\displaystyle a_n=a_1+(n-1)r\ ........[eq\ 1][/tex]
And the sum of all n terms is
[tex]\displaystyle s_n=\frac{a_1+a_n}{2}n...... [eq\ 2][/tex]
The sequence of the question complies with
[tex]\displaystyle s_n=342[/tex]
[tex]\displaystyle a_n=3a_1[/tex]
Using the last condition in eq 1 and knowing that r=1 (consecutive numbers)
[tex]\displaystyle a_n=a_1+n-1=3a_1[/tex]
Rearranging
[tex]\displaystyle 2a_1=n-1[/tex]
Using eq 2
[tex]\displaystyle \frac{a_1+a_n}{2}n=342[/tex]
Replacing the first condition
[tex]\displaystyle \frac{a_1+3a_1}{2}n=342[/tex]
Simplifying
[tex]\displaystyle 2a_1\ n=342[/tex]
Since
[tex]\displaystyle 2a_1=n-1[/tex]
We have
[tex]\displaystyle n(n-1)=342[/tex]
Factoring
[tex]\displaystyle n(n-1)=(19)(18)[/tex]
We find the number of terms
[tex]\displaystyle n=19[/tex]
The first term is
[tex]\displaystyle a_1=\ \frac{342}{38}=9[/tex]
Final answer:
The smallest integer is 6, and the sequence contains 19 terms.
Explanation:
To solve the problem about a sequence of consecutive integers where the sum is 342 and the largest integer is three times the smallest integer, we will use the formula for the sum of an arithmetic sequence and set up a system of equations. The sum of an arithmetic sequence is given by: S = ½ n(first integer + last integer), where S is the sum of the sequence, n is the number of terms, the first integer is a, and the last integer is l. We are given S = 342 and l = 3a.
Let's set up the system of equations:
S = ½ n(a + l)l = 3aS = 342By substituting l = 3a into the first equation, we get:
342 = ½ n(a + 3a)342 = ½ n(4a)Hence, n and a must be factors of 684 (since 342 = 2 × 171 = 4 × 342). Through trial and error or using a system of linear equations, we can find the appropriate values of n and a that will satisfy both the sum and the relationship between the smallest and largest integers.
Ultimately, we find that the smallest integer in the sequence is 6, and the sequence contains 19 terms.
Which expression is the best estimate of the product of 7/8 and 8 1/10?
A. 0x8
B. 1x10
C. 7x8
D. 1x8
Answer:
Option D -[tex]\frac{7}{8}\times 8\frac{1}{10}\approx 1\times 8[/tex]
Step-by-step explanation:
To find : Which expression is the best estimate of the product of [tex]\frac{7}{8}[/tex] and [tex]8\frac{1}{10}[/tex]?
Solution :
We estimate the number individually,
[tex]\frac{7}{8}=0.875[/tex]
Estimate the number we get 0.875≈1.
[tex]8\frac{1}{10}=\frac{81}{10}[/tex]
[tex]8\frac{1}{10}=8.1[/tex]
Estimate the number we get 8.1≈8.
The product of [tex]\frac{7}{8}[/tex] and [tex]8\frac{1}{10}[/tex] is
[tex]\frac{7}{8}\times 8\frac{1}{10}\approx 1\times 8[/tex]
Therefore, option D is correct.
A rancher purchased an SUV for $33,714 and made a down payment of 15% of the cost. The balance was financed for 4 years at an annual interest rate of 7%. Find the monthly truck payment.
Formula for monthly payment is:
A = P x (r(1+r)^t)/((1+r)^t-1) where P is the amount financed, r is the interest rate divided by 12 and t is the amount of time for the loan in months.
P = 33714 x 0.85 = 28656.90
A = 28656.90 x (0.07/12 (1+0.07/12)^48) / (1 +0.07/12)^48 - 1)
A = $686.23
A hemispherical plate with diameter 6 ft is submerged vertically 1 ft below the surface of the water. Express the hydrostatic force against one side of the plate as an integral and evaluate it. (Round your answer to the nearest whole number. Recall that the weight density of water is 62.5 lb/ft3.) 2δ 3 Correct: Your answer is correct. 0 dy ≈ lb
Answer:
F = 7476 N
Step-by-step explanation:
given,
diameter of hemispherical plate = 6 ft
height of submergence = 1 ft
the weight density of water = 62.5 lb/ft³
Assuming that hemispherical plate is residing on x and y axis.
bottom of plate is on x-axis and left side of the plate touches y-axis
now, plate is defined by the upper half of the circle
(x - 3)² + (y-0)² = 3²
y² = 9 - (x - 3)²
y = √(9 - (x - 3)²)
hydro static pressure on one side of plate.
[tex]F = \int \rho g x w(x)dx[/tex]
[tex]F = \int_0^3 62.5\times 9.8 x \times \sqrt{9-(x-3)^2}dx[/tex]
[tex]F = 612.5 \int_0^3 x \times \sqrt{9-(x-3)^2}dx[/tex]
on solving the above equation
[tex]F = 612.5(27\dfrac{\pi}{4}-9)[/tex]
F = 7476 N
Use the Divergence Theorem to compute the net outward flux of the following field across the given surface S.
F = < 8y^2 - 3x, -9x+4y, -2y^3 +z >
S is the sphere {(x,y,z): x^2 + y^2 + z^2 = 9}
Find net outward flux across the surface.
Answer:
Flux across S = 72π
Step-by-step explanation:
First we need to calculate the divergence of the vector field:
Div F = [tex]\frac{dFx}{dx} + \frac{dfy}{dy} + \frac{dFz}{dz}[/tex]
Where
Fx = 8y^2 - 3x
Fy = -9x+4y
Fz = -2y^3 +z
Then
Div F = -3 +4 + 1 = 2
And how the vector field’s divergence is a constant, we can calculate the flux across of the surface how:
Flux across S = Div F * Volume of Sphere
Fluz acroos S = 2(4/3)π[tex]r^{3}[/tex]
r : Sphere’s radio
Flux across S = (2)(4/3)π[tex]3^{3}[/tex]
= 72π
To compute the net outward flux of the field across the sphere, first calculate the divergence of the field and then apply the Divergence Theorem. The flux equals the integral of the divergence over the volume of the sphere.
Explanation:The first step of this problem is to compute the divergence of the vector field F. The divergence is the scalar quantity obtained by performing a dot product of the del operator with the field. For the given field F = <8y² - 3x, -9x + 4y, -2y³+ z>, the divergence is thus Div(F) = ∇.F = d(8y² - 3x)/dx + d(-9x + 4y)/dy + d(-2y³ + z)/dz.
For the sphere S where x² + y² + z² = 9, the radius r is √9 = 3. According to the Divergence Theorem, the flux across the boundary of the volume enclosed by S equals the triple integral of the divergence over the volume. So, the outward flux = ∫∫∫(Div(F).dV), where the triple integral is taken over the volume of the sphere.
Carry out the calculations to find the exact value of the outward flux.
Learn more about Divergence Theorem here:https://brainly.com/question/31272239
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A taxi driver had 44 fares to and from the airport last Monday. The price for a ride to the airport is $7, and the price for a ride from the airport is $6. The driver collected a total of $289 for the day.
Let x represent the number of trips to the airport and y represent the number of trips from the airport. Write the ordered pair (x,y) that represents the solution in this situation.
Your Answer Should Be
3m+6=24
Answer:
Step-by-step explanation:
Let x represent the number of trips to the airport and
Let y represent the number of trips from the airport.
A taxi driver had 44 fares to and from the airport last Monday. This means that
x + y = 44
The price for a ride to the airport is $7, and the price for a ride from the airport is $6. The driver collected a total of $289 for the day. This means that
7x + 6y = 289 - - - - - - - - - - - 1
Substituting x = 44 - y into equation 1, it becomes
7(44 - y) + 6y = 289
308 - 7y + 6y = 289
- 7y + 6y = 289 - 308
-y = - 19
y = 19
x = 44 - y
x = 44 - 19
x = 25
Classify the following differential equation: dy dx = y(y − 2)e x
i. ORDER:
ii. LINEAR/NONLINEAR:
iii. SEPARABLE/NOT SEPARABLE:
Answer:
1,non linear and separable
Step-by-step explanation:
given is a differential equation as
[tex]\frac{dy}{dx} =y(y-2)e^x[/tex]
Here we have derivative highest is first derivative
I) Order = 1 (since first derivative is used)
2) It is not linear since the variable y has power 2.
3) To check whether separable or not
[tex]\frac{dy}{dx} =y(y-2)e^x[/tex]
we can take all y variables to left side and x to right side
Hence separable
The answer is 16, I am just not sure how to arrive at that answer.
Step-by-step explanation:
∑ (4ⁿ⁺¹ / 5ⁿ)
Rewrite 4ⁿ⁺¹ as 4 (4ⁿ).
∑ 4 (4ⁿ / 5ⁿ)
∑ 4 (4/5)ⁿ
This is a geometric series. The sum of an infinite geometric series is:
S = a / (1 −r)
where a is the first term and r is the common ratio.
Here, the first term is 16/5 (because n starts at 1), and the common ratio is 4/5.
S = 16/5 / (1−4/5)
S = 16/5 / (1/5)
S = 16
If a couple were planning to have three children, the sample space summarizing the gender outcomes would be: bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg.A.) construct a similiar sample space for the possible weight outcomes (using o for overweight and u for underweight) of two children.B.) Assuming that the outcomes listed in part (a) were equally likely, find the probablity of getting two underweight children.C.) Find the probability of getting one overweight child and one underweight child.
Answer:
(A) oo, ou, uo, uu
(B) 1/4
(C) 1/2
Step-by-step explanation:
(A) When using o for overweight and u for underweight, there are four possible weight outcomes which are; oo, ou, uo, uu
The sample space would be: oo, ou, uo, uu
This implies there are 4 possible outcomes.
(B) From the sample space, the event, getting two underweight weight children occurs only once, uu. The probability of getting two underweight children = 1/4
(C) From the sample space, the event, getting one overweight child and one underweight child occurs twice, ou, uo.
The probability of getting one overweight child and one underweight child = 2/4 = 1/2
An insurance company is interested in conducting a study to to estimate the population proportion of teenagers who obtain a driving permit within 1 year of their 16th birthday. A level of confidence of 99% will be used and an error of no more than .04 is desired. There is no knowledge as to what the population proportion will be. The size of sample should be at least _______.
Answer:
n=1041 or higher
Step-by-step explanation:
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The population proportion have the following distribution
[tex]p \sim N(p,\sqrt{\frac{\hat p(1-\hat p)}{n}})[/tex]
2) Solution to the problem
In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 99% of confidence, our significance level would be given by [tex]\alpha=1-0.99=0.01[/tex] and [tex]\alpha/2 =0.005[/tex]. And the critical value would be given by:
[tex]z_{\alpha/2}=-2.58, z_{1-\alpha/2}=2.58[/tex]
The margin of error for the proportion interval is given by this formula:
[tex] ME=z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex] (a)
And on this case we have that [tex]ME =\pm 0.04[/tex] and we are interested in order to find the value of n, if we solve n from equation (a) we got:
[tex]n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2}[/tex] (b)
Since we don't have a prior estimation for th proportion of interest, we can use this value as an estimation [tex]\hat p =0.5[/tex] And replacing into equation (b) the values from part a we got:
[tex]n=\frac{0.5(1-0.5)}{(\frac{0.04}{2.58})^2}=1040.06[/tex]
And rounded up we have that n=1041 or higher.
Say you're playing three-card poker; that is, you're dealt three cards in a row at random from a standard deck of 52 cards. What are the odds of getting a pair or three of a kind?
Answer:
Step-by-step explanation:
Given
There are 52 cards in total
there are total of 13 pairs of same cards with each pair containing 4 cards
Probability of getting a pair or three of kind card=1-Probability of all three cards being different
Probability of selecting all three different cards can be find out by selecting a card from first 13 pairs and remaining 2 cards from remaining 12 pairs i.e.
[tex]=\frac{52\times 48\times 44}{52\times 51\times 50}[/tex]
for first card there are 52 options after choosing first card one pair is destroyed as we have to select different card .
For second card we have to select from remaining 12 pairs i.e. 48 cards and so on for third card.
Required Probability is [tex]=1-\frac{52\times 48\times 44}{52\times 51\times 50}[/tex]
[tex]=\frac{22776}{132600}[/tex]
Ron finds 9 books at a bookstore that he would like to buy, but he can afford only 5 of them. In how many ways can he make his selection? How many ways can he make his selection if he decides that one of the books is a must?
a. 3024; 1680
b. 7560; 840
c. 120; 1680
d. 126; 70
Answer:
d. 126; 70
Step-by-step explanation:
given that Ron finds 9 books at a bookstore that he would like to buy, but he can afford only 5 of them.
Out of 9 books he has to select 5 of them
Here selection of books can be done in any order. Hence order does not matter
No of ways he selects 5 books out of 9 books = 9C5
= 126
Part II
One book is a must
Hence he has options only 4 books from the remaining 8.
No of ways when one book is compulsory = selecting 3 books from 8 books
= 8C4
= 70
Option d is right.
a taxi company charges passengers $1.00 for a ride, and an additional $0.30
A rock thrown vertically upward from the surface of the moon at a velocity of 36m/sec reaches a height of s = 36t - 0.8 t^2 meters in t sec.
a. Find the rock's velocity and acceleration at time t.
b. How long does it take the rock to reach its highest point?
c. How high does the rock go?
d. How long does it take the rock to reach half its maximum height?
e. How long is the rock a loft?
Answer:
a. The rock's velocity is [tex]v(t)=36-1.6t \:{(m/s)}[/tex] and the acceleration is [tex]a(t)=-1.6 \:{(m/s^2)}[/tex]
b. It takes 22.5 seconds to reach the highest point.
c. The rock goes up to 405 m.
d. It reach half its maximum height when time is 6.59 s or 38.41 s.
e. The rock is aloft for 45 seconds.
Step-by-step explanation:
Velocity is defined as the rate of change of position or the rate of displacement. [tex]v(t)=\frac{ds}{dt}[/tex]Acceleration is defined as the rate of change of velocity. [tex]a(t)=\frac{dv}{dt}[/tex]a.
The rock's velocity is the derivative of the height function [tex]s(t) = 36t - 0.8 t^2[/tex]
[tex]v(t)=\frac{d}{dt}(36t - 0.8 t^2) \\\\\mathrm{Apply\:the\:Sum/Difference\:Rule}:\quad \left(f\pm g\right)'=f\:'\pm g'\\\\v(t)=\frac{d}{dt}\left(36t\right)-\frac{d}{dt}\left(0.8t^2\right)\\\\v(t)=36-1.6t[/tex]
The rock's acceleration is the derivative of the velocity function [tex]v(t)=36-1.6t[/tex]
[tex]a(t)=\frac{d}{dt}(36-1.6t)\\\\a(t)=-1.6[/tex]
b. The rock will reach its highest point when the velocity becomes zero.
[tex]v(t)=36-1.6t=0\\36\cdot \:10-1.6t\cdot \:10=0\cdot \:10\\360-16t=0\\360-16t-360=0-360\\-16t=-360\\t=\frac{45}{2}=22.5[/tex]
It takes 22.5 seconds to reach the highest point.
c. The rock reach its highest point when t = 22.5 s
Thus
[tex]s(22.5) = 36(22.5) - 0.8 (22.5)^2\\s(22.5) =405[/tex]
So the rock goes up to 405 m.
d. The maximum height is 405 m. So the half of its maximum height = [tex] \frac{405}{2} =202.5 \:m[/tex]
To find the time it reach half its maximum height, we need to solve
[tex]36t - 0.8 t^2=202.5\\36t\cdot \:10-0.8t^2\cdot \:10=202.5\cdot \:10\\360t-8t^2=2025\\360t-8t^2-2025=2025-2025\\-8t^2+360t-2025=0[/tex]
For a quadratic equation of the form [tex]ax^2+bx+c=0[/tex] the solutions are
[tex]x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
[tex]\mathrm{For\:}\quad a=-8,\:b=360,\:c=-2025:\\\\t=\frac{-360+\sqrt{360^2-4\left(-8\right)\left(-2025\right)}}{2\left(-8\right)}=\frac{45\left(2-\sqrt{2}\right)}{4}\approx 6.59\\\\t=\frac{-360-\sqrt{360^2-4\left(-8\right)\left(-2025\right)}}{2\left(-8\right)}=\frac{45\left(2+\sqrt{2}\right)}{4}\approx 38.41[/tex]
It reach half its maximum height when time is 6.59 s or 38.41 s.
e. It is aloft until s(t) = 0 again
[tex]36t - 0.8 t^2=0\\\\\mathrm{Factor\:}36t-0.8t^2\rightarrow -t\left(0.8t-36\right)\\\\\mathrm{The\:solutions\:to\:the\:quadratic\:equation\:are:}\\\\t=0,\:t=45[/tex]
The rock is aloft for 45 seconds.
Final answer:
The question explores kinematic principles by calculating the velocity, acceleration, time to reach the highest point, maximum height, time to reach half the maximum height, and total duration aloft for a rock thrown vertically on the moon, giving us the values as follows upon solving.
v(t) = (36 - 1.6t) m/s
a(t) = -1.6 m/s²
t_highest = 22.5 seconds
h_max = 405 meters
t_half ≈ 11.6 seconds
t_total = 45 seconds
Explanation:
The question involves calculating velocity, acceleration, and the dynamics of a rock thrown vertically on the moon, illustrating concepts from kinematic equations in physics.
a. Velocity and acceleration at time t
The velocity of the rock at time t is given by the derivative of the position function s = 36t - 0.8t², which is v(t) = 36 - 1.6t Acceleration, being the derivative of velocity, is constant at -1.6 m/s², due to moon's gravity.
b. Time to reach the highest point
At the highest point, the velocity is 0. Setting v(t) = 0, we find t = 22.5 seconds.
c. Height at the highest point
Plugging t = 22.5 into the position function, we find the maximum height is 405 meters.
d. Time to reach half the maximum height
Setting s = 202.5 meters in the original equation and solving for t, we find two values, but the relevant one is approximately 11.6 seconds.
e. Total duration aloft
To find when the rock returns to the surface, set s = 0 in the original equation and solve for t, giving a total duration of 45 seconds.
If X and Y are any random variables with E(X) = 5, E(Y) = 6, E(XY) = 21, V(X) = 9 and V(Y) = 10, then the relationship between X and Y is a:
-strong positive relationship
-strong negative relationship
-weak positive relationship
-weak negative relationship
Answer:
We have a strong negative relationship between the variables.
Step-by-step explanation:
Given two random variables X and Y, it is possible to calculate the covariance as Cov(X, Y) = E(XY)-E(X)E(Y). We have E(X)=5, E(Y)=6 and E(XY)=21. Therefore Cov(X,Y)=21-(5)(6)=21-30=-9. On the other hand, we know that the correlation of X and Y is the number defined by [tex]Cov(X,Y)/\sqrt{Var(X)}\sqrt{Var(Y)}[/tex] and because in this particular case we have V(X)=9 and V(Y)=10, we have [tex]-9/\sqrt{9}\sqrt{10}[/tex] = -0.9487. Therefore, we have a strong negative relationship between the variables.
Final answer:
The X and Y variables have a strong negative relationship.
Explanation:
The X and Y variables have a strong negative relationship. This can be determined by analyzing the correlation coefficient, which indicates the strength and direction of the relationship between two variables.
In this case, since the correlation coefficient is significantly different from zero (positive or negative), we can conclude that there is a significant linear relationship between X and Y. The fact that the correlation coefficient is negative indicates that as X increases, Y tends to decrease, and vice versa.
Therefore, the correct answer is strong negative relationship.