State the postulate or theorem you would use to prove each pair of triangles congruent.
If the triangles cannot be proved congruent, choose "Not Possible."

Given the width and length of the rectangle in terms of x, the formula for the perimeter is substituted with these expressions. After simplifying, the perimeter of the rectangle is found to be represented by the equation 8.5x + 3.

The subject represents a problem in mathematics, specifically geometry. The perimeter of a rectangle is calculated by the formula 2(width + length). Given that the width is meant to be represented by (3x - 1.5) and the length is represented by (1.25x + 3), we substitute these expressions into our formula to find the perimeter of the rectangle.

Perimeter = 2[(3x - 1.5) + (1.25x + 3)].

To simplify, the above equation becomes:

Perimeter = 2[4.25x + 1.5],

which then further simplifies to:

Perimeter = 8.5x + 3.

The perimeter of the rectangle is therefore represented by the expression 8.5x + 3.

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Answer:

the nth term of the sequence is [tex]a_n=6n+7[/tex]

Step-by-step explanation:

Given : An arithmetic sequence begins as follows: [tex]a_1=13, a_2=19[/tex]

To find : Which of the following gives the definition of its nth term?

Solution :

The nth term of the A.P is [tex]a_n=a+(n-1)d[/tex]

The first term is [tex]a=a_1=13[/tex]

The common difference is [tex]d=a_2-a_1[/tex]

[tex]d=19-13=6[/tex]

Substitute in the formula,

[tex]a_n=13+(n-1)6[/tex]

[tex]a_n=13+6n-6[/tex]

[tex]a_n=6n+7[/tex]

Therefore, the nth term of the sequence is [tex]a_n=6n+7[/tex]

The nth term of the given arithmetic sequence is defined by the formula an = 13 + (n-1)*6. This is derived from the general formula for an arithmetic sequence and using the given first two terms.

In this given problem, we have an arithmetic sequence. An arithmetic sequence is a list of numbers in which each term is obtained by adding a constant difference to the preceding term. For this particular sequence, the first term (a1) is 13, and the second term (a2) is 19. Hence, the common difference (d) between the terms is 19 - 13 = 6.

The general formula for the nth term (an) of an arithmetic sequence is given by an = a1 + (n-1)*d. In this situation, to represent any term in the series, the formula would be an = 13 + (n-1)*6. This formula can generate any term in the sequence, given the term number.

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Answer:

3 necklaces.

Step-by-step explanation:

According to the Question,

Elise is having 15 Purple beads and 12 Yellow beads ,

All necklace should be the same ,

So, She wants to form group such that in each group , equal number of Purple and Yellow beads will be present.

Now , the maximum number of such necklaces can be calculated from the HCF of 15 and 12 as we want to break them in equal groups .

Now , HCF of 15 and 12 comes out to be 3.

Thus ,

a maximum of 3 necklaces can be made.

Answer:

130000m^2

Step-by-step explanation:

a = 800m

b = 800m

c= 650m

α = 30°

4th he triangular trench is an isosceles triangle.

Area of a triangle = 1/2(bcsinA)

= 1/2(800*650*sin30°)

= 130,000m^2

Answer:

The area enclosed by the trenches is 130 000[tex]m^{2}[/tex]

Step-by-step explanation:

The two given sides of the trenches are 800m and 650m. Since the included angle of the two sides are given, then the area covered by the trenches can be calculated using the formula;

     Area of a triangle = [tex]\frac{1}{2}[/tex] × abSin C

where: a is the length of one side and b the length of the second side and C represents the value of the included angle.

Thus,

    a = 800m, b = 650m and C = 30°

So that,

Area enclosed by the trenches  = [tex]\frac{1}{2}[/tex] × abSin C

                                                          =  [tex]\frac{1}{2}[/tex] × 800 × 650 × Sin30°

                                                          = [tex]\frac{1}{2}[/tex] × 800 × 650 × 0.5

                                                          = 130 000

The area enclosed by the trenches  is 130 000[tex]m^{2}[/tex].

Answer:

$150

Step-by-step explanation:

Dad gives the girl 50 and she save 20 so 50-20 = 30. Then u multiply that by the number of months given...so 30*5= 150

Answer: 150

Step-by-step explanation:

50-20=30

30x5=150

Answer:

  "closed"

Step-by-step explanation:

A set is closed with respect to an operation if the operation on members of the set produces a member of the set.

Answer: D. [tex]\dfrac{qd}{(q+2)}[/tex]

Step-by-step explanation:

Given : q workers can paint a house in d days.

Let [tex]d_1[/tex] be the number of days taken by q+2 workers to paint the same house.

Since there is inverse relationship between the number of workers and the number of days to do same work ( condition - all workers paints at the same rate), as the number of workers increases the number of days to complete it decreases.

Equation of inverse variation between x and y  : [tex]x_1y_1=x_2y_2[/tex]

Substitute , [tex]x_1=q ,\ y_1=d[/tex] and  [tex]x_2=q+2 ,\ y_2=d_2[/tex] , we get

[tex]qd=(q+2)d_2\\\\\Rightarrow\ d_2=\dfrac{qd}{(q+2)}[/tex]

Therefore , the number of days it will take  q+2 workers to paint the same house = [tex]\dfrac{qd}{(q+2)}[/tex]

Hence, the correct answer is : D. [tex]\dfrac{qd}{(q+2)}[/tex]

Answer:

Solution

(x, y) = (2, 3)

Step-by-step explanation:

Answer:

Option 3)

[tex]x^3-9^3 = (x-9)(x^2 + 9x + 81)[/tex]

Step-by-step explanation:

We use the identities:

[tex]a^3 + b^3 = (a+b)(a^2-ab+b^2)\\a^3-b^3 = (a-b)(a^2+ab+b^2)[/tex]

1.

[tex](x + 7)(x^2 -7x + 14)[/tex]

It is not a sum or difference of cubes because it does not satisfies the identity.

2.

[tex](x + 8)(x^2 + 8x + 64)[/tex]

It is not a sum or difference of cubes because it does not satisfies the identity.

3.

[tex](x - 9)(x^2 + 9x + 81)\\\text{Comparing with the identity:} \\a^3-b^3 = (a-b)(a^2+ab+b^2)\\\text{We get}\\a = x\\b = 9\\x^3-9^3 = (x-9)(x^2 + 9x + 81)[/tex]

Thus, it can be expressed as a difference of cubes.

4.

[tex](x - 10)(x^2 - 10x + 100)[/tex]

It is not a sum or difference of cubes because it does not satisfies the identity.

Answer:

C

Step-by-step explanation:

Answer: the number of the parking space covered by the car is 87

Step-by-step explanation:

Numbers are assigned to each parking spot. Looking closely at the numbers assigned to each spot, the numbers are inverted and the number on each successive spot differ by one. The numbers are 86, 87, 88, 89, 90, 91

Therefore, the number assigned to the spot where the car would be 87

Answer:

[tex] x= 5*400 ft = 2000 ft[/tex]

Step-by-step explanation:

For this case we can use the figure attached and we are interested in order to find the value of x.

We have two similar triangles (DEC and ABC) and we can find the scale factor like this:

[tex]Factor= \frac{EC}{BC}=\frac{500ft}{100ft}=5[/tex]

And now we can apply proportions in order to find the value of x using the two sides DE and BA, since we have the ratio between the triangle DEC and ABC we have this:

[tex]Factor=5=\frac{x ft}{400 ft}[/tex]

And solving for x we got:

[tex] x= 5*400 ft = 2000 ft[/tex]

And then the distance across the lake would be 2000 ft

Answer:

ewq

Step-by-step explanation:

Step-by-step explanation:

9.5 × 10^(5) is greater than 1.2 × 10^(4) is greater than 8.1 × 10^(-6) is greater than -7.2×10^(-4)

D > C > B > A

To order the numbers from least to greatest, one must compare the exponents in scientific notation. The correct sequence from least to greatest is B, A, C, D.

The question asks to order the given numbers from least to greatest. The numbers are provided in scientific notation, which is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. To compare these numbers, we look at their power of 10, as this indicates the number's scale or size.

'A' is 7.2*10^-4 which is a small positive number.

'B' is 8.1*10^-6 which is even smaller, given the more negative exponent.

'C' is 1.2*10^4 which is a large positive number.

'D' is 9.5*10^5 which is larger than 'C', given the larger exponent.

Therefore, when ordered from least to greatest, the sequence is: B, A, C, D.

The function that represents the number of E.coli bacteria cells per 100 mL of water as t years elapses, and is missing in the question, is:

[tex]A(t)=136(1.123)^{4t}[/tex]

Answer:

   Option A. 59%

Explanation:

The base, 1.123, represents the multiplicative constant rate of change of the function, so you just must substitute 1 for t in the power part of the function::

Then, the multiplicative rate of change is 1.590, which means that every year the number of E.coli bacteria cells per 100 mL of water increases by a factor of 1.590, and that is 1.59 - 1 = 0.590 or 59% increase.

You can calculate it also using two consecutive values for t. For instance, use t =1 and t = 1

[tex]t=1\\\\ A(1)=136(1.123)^4\\\\\\t=2\\ \\ \\A(2)=136(1.123)^8\\ \\ \\A(2)/A(1)=1.123^8/1.1123^4=1.123^4=1.590[/tex]

Answer:

A. 2.8 minutes

Step-by-step explanation:

1) Previous concepts

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 Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

2) Solution to the problem

Let X the random variable that represent the length of time it takes a college student to find a parking spot in the library parking of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(3.5,1)[/tex]  

Where [tex]\mu=3.5[/tex] and [tex]\sigma=1[/tex]

And we need to find a value a, such that we satisfy this condition:

[tex]P(X>a)=0.758[/tex]   (a)

[tex]P(X<a)=0.242[/tex]   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.242 of the area on the left and 0.758 of the area on the right it's z=-0.700. On this case P(Z<-0.700)=0.242 and P(z>-0.700)=0.758

If we use condition (b) from previous we have this:

[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.242[/tex]  

[tex]P(z<\frac{a-\mu}{\sigma})=0.242[/tex]

But we know which value of z satisfy the previous equation so then we can do this:

[tex]z=-0.700<\frac{a-3.5}{1}[/tex]

And if we solve for a we got

[tex]a=3.5 -0.700*1=2.8[/tex]

So the value of height that separates the bottom 24.2% of data from the top 75.8% is 2.8 minutes.  

The point in time where 75.8% of students exceed when finding a parking spot is calculated using Z-scores in a normal distribution. The time is approximately 2.8 minutes.

The process to find the point in the distribution where 75.8% of the students exceed the required time to find a parking spot in the library parking lot is essentially to calculate the Z-score corresponding to a given cumulative probability (or area) in the standard normal curve.  If the area to the left of the Z-score is 75.8%, then the area to the right of the Z-score is 100%-75.8% = 24.2%.

We then find the Z-score corresponding to 24.2% in the Z-tables. In this case, the corresponding Z-score is approximately -0.7 (since the tables usually give the area to the left, and we are dealing with the area to the right of the Z-score, we consider the negative of the found Z-score).

To convert this back to the time it takes to find a parking spot, we use the formula for converting Z-scores back to raw scores: X = μ + Zσ, where X is the raw score (time to find a parking spot), μ is the mean time to find a parking spot, Z is the Z-score, and σ is the standard deviation of the time to find a parking spot.

Substituting the given values, we get X = 3.5 + (-0.7)*1 = 2.8.

Thus, the time at which 75.8% of the college students exceed when trying to find a parking spot in the library parking lot is 2.8 minutes (option A).

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Answer:

[tex]p_v = P(\chi^2_{4,0.05} >3.81)=0.43233[/tex]

Since the p values is higher than the significance level we FAIL to reject the null hypothesis at 5% of significance, and we can conclude that we don't have significant differences between the 3 remedies analyzed. So we can say that the 3 remedies ar approximately equally effective.

Step-by-step explanation:

A chi-square goodness of fit test "determines if a sample data matches a population".

A chi-square test for independence "compares two variables in a contingency table to see if they are related. In a more general sense, it tests to see whether distributions of categorical variables differ from each another".

Assume the following dataset:

                               NyQuil          Robitussin       Triaminic     Total

No relief                    11                     13                      9               33

Some relief               32                   28                     27              87

Total relief                7                       9                      14               30

Total                         50                    50                    50              150

We need to conduct a chi square test in order to check the following hypothesis:

H0: There is no difference in the three remedies

H1: There is a difference in the three remedies

The level os significance assumed for this case is [tex]\alpha=0.05[/tex]

The statistic to check the hypothesis is given by:

[tex]\sum_{i=1}^n \frac{(O_i -E_i)^2}{E_i}[/tex]

The table given represent the observed values, we just need to calculate the expected values with the following formula [tex]E_i = \frac{total col * total row}{grand total}[/tex]

And the calculations are given by:

[tex]E_{1} =\frac{50*33}{150}=11[/tex]

[tex]E_{2} =\frac{50*33}{150}=11[/tex]

[tex]E_{3} =\frac{50*33}{150}=11[/tex]

[tex]E_{4} =\frac{50*87}{150}=29[/tex]

[tex]E_{5} =\frac{50*87}{150}=29[/tex]

[tex]E_{6} =\frac{50*87}{150}=29[/tex]

[tex]E_{7} =\frac{50*30}{150}=10[/tex]

[tex]E_{8} =\frac{50*30}{150}=10[/tex]

[tex]E_{9} =\frac{50*30}{150}=10[/tex]

And the expected values are given by:

                               NyQuil          Robitussin       Triaminic     Total

No relief                    11                     11                       11               33

Some relief               29                   29                     29              87

Total relief                10                     10                     10               30

Total                         50                    50                    50              150

And now we can calculate the statistic:

[tex]\chi^2 = \frac{(11-11)^2}{11}+\frac{(13-11)^2}{11}+\frac{(9-11)^2}{11}+\frac{(32-29)^2}{29}+\frac{(28-29)^2}{29}+\frac{(27-29)^2}{29}+\frac{(7-10)^2}{10}+\frac{(9-10)^2}{10}+\frac{(14-10)^2}{10} =3.81[/tex]

Now we can calculate the degrees of freedom for the statistic given by:

[tex]df=(rows-1)(cols-1)=(3-1)(3-1)=4[/tex]

And we can calculate the p value given by:

[tex]p_v = P(\chi^2_{4,0.05} >3.81)=0.43233[/tex]

And we can find the p value using the following excel code:

"=1-CHISQ.DIST(3.81,4,TRUE)"

Since the p values is higher than the significance level we FAIL to reject the null hypothesis at 5% of significance, and we can conclude that we don't have significant differences between the 3 remedies analyzed.

To test the hypothesis that three cough remedies are equally effective, conduct an Analysis of Variance (ANOVA) test and use the p-value, compared to the pre-set level of 0.05, to decide if the null hypothesis should be rejected or failed to reject.

The subject matter pertains to hypothesis testing, a method used in statistics to test the validity of a claim (hypothesis) about a population. In this case, the null hypothesis in your question is that the three cough remedies are equally effective. The alternative hypothesis (Ha) is that at least one of the remedies is different. The P-value, when set at 0.05, helps to decide on the rejection or non-rejection of the null hypothesis, depending on whether it’s greater or smaller than the P-value.

First, we need to perform an Analysis of Variance (ANOVA) test since we have more than two samples to compare. After conducting the test, we compare the P-value with our pre-set alpha (0.05). If the p-value obtained is less than 0.05, we can reject the null hypothesis, letting us conclude that not all cough remedies are equally effective. If, however, the p-value is more than 0.05, we fail to reject the null hypothesis, and hence, we can't confidently say that one cough remedy is more effective than the others.

Remember that failing to reject the null hypothesis does not prove it true, it only suggests that there's not enough evidence against it given our data and chosen significance level.

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Answer:

View Image

Step-by-step explanation:

Solve for y.

You have a ≥ so it's a solid line and you shade above that line.

Final answer:

The point in Quadrant I with an x-value 2 units from the origin and a y-value 7 units from the origin has coordinates (2, 7).

Explanation:

The student is asked to determine the coordinates of a point that lies in Quadrant I of the Cartesian coordinate system, with an x-value that is 2 units from the origin and a y-value 7 units from the origin.

The Cartesian coordinate system is divided into four quadrants, and Quadrant I is where both x and y values are positive.

Therefore, a point located 2 units away from the origin on the x-axis, and 7 units away on the y-axis, would have coordinates (2, 7).

Remember, the origin is the point where the x-axis and y-axis intersect, (0,0). Moving 2 units to the right, you reach an x-value of 2. Then, moving up 7 units, you achieve a y-value of 7.

Thus, the coordinates are (2, 7).

The function's derivative, f'(x) = x²(x + 1)³3(x - 4)³, has 2 relative maximum and minimum points.

So, the correct answer is C) 2.

The function's derivative, f'(x) =  x²(x + 1)³3(x - 4)³, gives us information about the critical points of the function. Relative maximum and minimum points occur where the derivative is zero or undefined. To find these points, we set the derivative equal to zero and solve for x:  x²(x + 1)³3(x - 4)³ = 0. By analyzing the signs of the factors, we can determine the number of relative maximum and minimum points:

Therefore, the total number of relative maximum and minimum points is 2.

So, the correct answer is C) 2.

Answer:

[tex]x=\frac{83}{50}[/tex]

Step-by-step explanation:

we know that

If the three points are collinear

then

[tex]m_A_B=m_A_C[/tex]

we have

A (1, 2/3), B (x, -4/5), and C (-1/2, 4)

The formula to calculate the slope between two points is equal to

[tex]m=\frac{y2-y1}{x2-x1}[/tex]

step 1

Find the slope AB

we have

[tex]A(1,\frac{2}{3}),B(x,-\frac{4}{5})[/tex]

substitute in the formula

[tex]m_A_B=\frac{-\frac{4}{5}-\frac{2}{3}}{x-1}[/tex]

[tex]m_A_B=\frac{\frac{-12-10}{15}}{x-1}[/tex]

[tex]m_A_B=-\frac{22}{15(x-1)}[/tex]

step 2

Find the slope AC

we have

[tex]A(1,\frac{2}{3}),C(-\frac{1}{2},4)[/tex]

substitute in the formula

[tex]m_A_C=\frac{4-\frac{2}{3}}{-\frac{1}{2}-1}[/tex]

[tex]m_A_C=\frac{\frac{10}{3}}{-\frac{3}{2}}[/tex]

[tex]m_A_C=-\frac{20}{9}[/tex]

step 3

Equate the slopes

[tex]m_A_B=m_A_C[/tex]

[tex]-\frac{22}{15(x-1)}=-\frac{20}{9}[/tex]

solve for x

[tex]15(x-1)20=22(9)[/tex]

[tex]300x-300=198[/tex]

[tex]300x=198+300[/tex]

[tex]300x=498[/tex]

[tex]x=\frac{498}{300}[/tex]

simplify

[tex]x=\frac{83}{50}[/tex]

Answer:

??????????????????? what

Answer:

a) [tex]P(X=0)=(4C0)(0.75)^0 (1-0.75)^{4-0}=0.0039[/tex]  

[tex]P(X=1)=(4C1)(0.75)^1 (1-0.75)^{4-1}=0.0469[/tex]  

[tex]P(X=2)=(4C2)(0.75)^2 (1-0.75)^{4-2}=0.211[/tex]  

[tex]P(X=3)=(4C3)(0.75)^2 (1-0.75)^{4-3}=0.422[/tex]  

[tex]P(X=4)=(4C4)(0.75)^2 (1-0.75)^{4-4}=0.316[/tex]

b) [tex] E(X) = np = 4*0.75=3[/tex]

c) [tex] Sd(X) =\sqrt{np(1-p)}=\sqrt{4*0.75*(1-0.75)}=0.866[/tex]

d) [tex] P(X \geq 3) \geq 0.98[/tex]

And the dsitribution that satisfy this is [tex]X\sim Binom(n=9,p=0.75[/tex]

Step-by-step explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Let X the random variable of interest, on this case we now that:

[tex]X \sim Binom(n=4, p=1-0.25=0.75)[/tex]

The probability mass function for the Binomial distribution is given as:

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]

Where (nCx) means combinatory and it's given by this formula:

[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]

Part a

[tex]P(X=0)=(4C0)(0.75)^0 (1-0.75)^{4-0}=0.0039[/tex]  

[tex]P(X=1)=(4C1)(0.75)^1 (1-0.75)^{4-1}=0.0469[/tex]  

[tex]P(X=2)=(4C2)(0.75)^2 (1-0.75)^{4-2}=0.211[/tex]  

[tex]P(X=3)=(4C3)(0.75)^2 (1-0.75)^{4-3}=0.422[/tex]  

[tex]P(X=4)=(4C4)(0.75)^2 (1-0.75)^{4-4}=0.316[/tex]

Part b

The expected value is givn by:

[tex] E(X) = np = 4*0.75=3[/tex]

Part c

For the standard deviation we have this:

[tex]Sd(X) =\sqrt{np(1-p)}=\sqrt{4*0.75*(1-0.75)}=0.866[/tex]

Part d

For this case the sample size needs to be higher or equal to 9. Since we need a value such that:

[tex] P(X \geq 3) \geq 0.98[/tex]

And the dsitribution that satisfy this is [tex]X\sim Binom(n=9,p=0.75[/tex]

We can verify this using the following code:

"=1-BINOM.DIST(3,9,0.75,TRUE)" and we got 0.99 and the condition is satisfied.

To find the probabilities of different numbers of successes ranging from 0 to 4 in a group of parolees, we can use the binomial probability formula. The expected number of parolees who will not be repeat offenders can be calculated by multiplying the probability of success by the total number of parolees. The standard deviation can also be calculated using a formula. To be about 98% sure that three or more parolees will not become repeat offenders, Alice should counsel a group size of at least 13 parolees.

To find the probabilities of different numbers of successes, we can use the binomial probability formula. Given that the probability of success is 0.25 and the number of trials is 4, we can calculate the probabilities as follows:

The expected number of parolees who will not be repeat offenders can be calculated by multiplying the probability of success (0.25) by the total number of parolees (4), which gives an expected value of 1 parolee. The standard deviation can be calculated using the formula sqrt(n * p * (1-p)), where n is the number of trials and p is the probability of success. In this case, the standard deviation is sqrt(4 * 0.25 * (1-0.25)) ≈ 0.866.

To determine the size of the group that Alice should counsel to be about 98% sure that three or more parolees will not become repeat offenders, we can use the binomial cumulative distribution function. We need to find the smallest value of n (the number of trials) such that P(X ≥ 3) > 0.98, where X represents the number of successes. Solving this inequality, we find that Alice should counsel a group size of at least 13 parolees.

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the reason that these are inverses is because if you switch x with z

and g(f(x))=x, then (x-9)/(x+5) is (z-9)/(z+5)=x. which implies that:

(z+5) - 14/(z+5)=1-(14/(z-5))=x.

(1/(1 - x) * 14) + 5 = z = g(x) maybe i messed up?

Step-by-step explanation:

f(x) = (x − 9) / (x + 5)

g(x) = (-5x − 9) / (x − 1)

To find f(g(x)), substitute g(x) into f(x).

f(g(x)) = [(-5x − 9) / (x − 1) − 9] / [(-5x − 9) / (x − 1) + 5]

Multiply top and bottom by x − 1.

f(g(x)) = [(-5x − 9) − 9(x − 1)] / [(-5x − 9) + 5(x − 1)]

Simplify.

f(g(x)) = (-5x − 9 − 9x + 9) / (-5x − 9 + 5x − 5)

f(g(x)) = (-14x) / (-14)

f(g(x)) = x

To find g(f(x)), substitute f(x) into g(x).

g(f(x) = [-5(x − 9) / (x + 5) − 9] / [(x − 9) / (x + 5) − 1]

Multiply top and bottom by x + 5.

g(f(x) = [-5(x − 9) − 9(x + 5)] / [(x − 9) − (x + 5)]

Simplify.

g(f(x) = (-5x + 45 − 9x − 45) / (x − 9 − x − 5)

g(f(x) = (-14x) / (-14)

g(f(x) = x

Explanation:

a. The line joining the midpoints of the parallel bases is perpendicular to both of them. It is the line of symmetry for the trapezoid. This means the angles and sides on one side of that line of symmetry are congruent to the corresponding angles and sides on the other side of the line. The diagonals are the same length.

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b. We observe that adjacent pairs of points have the same x-coordinate, so are on vertical lines, which have undefined slope. KN is a segment of the line x=1; LM is a segment of the line x=3. If the trapezoid is isosceles, the midpoints of these segments will be on a horizontal line. The midpoint of KN is at y=(3-2)/2 = 1/2. The midpoint of LM is at y=(1+0)/2 = 1/2. These points are on the same horizontal line, so the trapezoid is isosceles.

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c. We observed in part (b) that the parallel sides are KN and LM. The coordinate difference between K and L is (1, 3) -(3, 1) = (-2, 2). That is, segment KL is the hypotenuse of an isosceles right triangle with side lengths 2, so the lengths of KL and MN are both 2√2.

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For part (c), we used the shortcut that the hypotenuse of an isosceles right triangle is √2 times the leg length.

Answer:

72 yrs old.

Step-by-step explanation:

The combined age

D+O=96

The owner is 3 times older than dog

O=3D

D+3D=96

4D = 96

D= 24

Now substitute the value of D in O=3D

O=3.24= 72

The owner is 72 years old.

To solve this problem, we can use algebra. Let's define the dog's age as 'd' and the owner's age as 'o'. By solving the two equations (o + d = 96 and o = 3d), we can determine that the dog is 24 years old and the owner is 72 years old.

The question asks us to find the age of a dog's owner, given that the combined ages of the owner and the dog are 96 years, and the owner's age is three times the age of the dog. We can use algebra to solve this problem.

Let's define the dog's age as 'd' and the owner's age as 'o'. We know that o = 3d (the owner's age is three times the dog's age) and o + d = 96 (the combined ages of the owner and the dog are 96).

To find the owner's age, substitute '3d' for 'o' in the second equation: 3d + d = 96. This simplifies to 4d = 96. Dividing both sides of the equation by 4 gives us d = 24, meaning the dog is 24 years old. Now we substitute d = 24 into the equation o = 3d, resulting in o = 72. Therefore, the owner is 72 years old.

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Step-by-step explanation:

the answer is

Y=2.5

X=3.75

Answer:

6 brownies and 2 cupcakes.

Step-by-step explanation:

Let the number of brownies be b and the number of cupckes be c.

We have the following system:

b = 3c

1.5b + 3c = 15

Substituting for  b in the second equation:

4.5c  + 3c = 15

7.5c = 15

c =  2.

Thus b = 3c = 6.

The coordinates provided represent vertices of a triangle on a 2-dimensional grid like a city map. Visualizing the points and connecting them creates a right triangle. Vectors can express the sides of this triangle in terms of magnitude and direction.

This is a problem pertaining to geometry and the use of coordinate systems. In the question, we're given vertices of a triangle on a coordinate grid. The coordinates, (1, 1), (1, 4), and (-3, 4), denote specific points on this two-dimensional grid system. Just as in a city map, these coordinates help define the exact location of the points in a well-defined and universally understood manner.

To visualize the triangle formed by these coordinates on a city map, we can consider (1, 1) as a starting point. From this point, move 3 units north to reach (1, 4), then move 4 units to the left to reach (-3, 4), and finally move back 3 units south to reach the starting point (1, 1). This forms a right triangle, with (1, 4) as the right-angle vertex.

Similar to the concept of directional travel mentioned in the supplementary text, vectors can also be utilized in such coordinate systems to describe movement or distance between different points on the map. In our triangle example, each side of the triangle can be represented by a vector, with its magnitude (length) and direction.

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To find the area of the triangle park, we can calculate the lengths of the sides using the distance formula and then use Heron's formula to find the area. The area of the triangle park is 6 square units.

The given question is about a city planner mapping out a triangle park on a grid paper using coordinates. The vertices of the park are given as (1, 1), (1, 4), and (-3, 4). The subject of this question is Mathematics, and the grade level is High School.

To find the area of the triangle park, we can use the formula for the area of a triangle. We connect the vertices with line segments and calculate the lengths of the sides. Then, using the Heron's formula, we can calculate the area.

First, we calculate the lengths of the sides using the distance formula. The length of AB is sqrt((1-1)^2 + (4-1)^2) = sqrt(9) = √9 = 3. The length of BC is sqrt((1-(-3))^2 + (4-4)^2) = sqrt(16) = √16 = 4. The length of AC is sqrt((1-(-3))^2 + (4-1)^2) = sqrt(25) = √25 = 5.

Next, we use Heron's formula to calculate the area. Heron's formula states that the area of a triangle with side lengths a, b, and c is given by: Area = sqrt(s(s-a)(s-b)(s-c)), where s is the semi-perimeter (s = (a+b+c)/2).

In this case, a = 3, b = 4, and c = 5. Therefore, s = (3+4+5)/2 = 12/2 = 6. Plugging in the values, the area of the triangle is given by: Area = sqrt(6(6-3)(6-4)(6-5)) = sqrt(6*3*2*1) = sqrt(36) = √36 = 6.

Therefore, the area of the triangle park is 6 square units.

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Answer:

Statements (1) and (2) TOGETHER are NOT sufficient.

Explanation:

As in the equation (327)(510)(z) = (58)(914)(xy) there are THREE variables in total i.e. "x", "y" and "z" hence minimum three equations are required to find out values of all variables. Hence,

If the given number of equations is equal to total variable used in any of the equation, values of all the variables can be find out otherwise there can be unlimited number of solutions.

So, value of "x" cannot be determined with the given data.

Answer:

  x 2 −2 3 −3

  y 4 4 9 −9

Step-by-step explanation:

A table does not represent a function if any x-value is repeated. The table shown above is the only one with unique x-values.

A linear function can be used to represent the total retail cost of renting a vacuum cleaner: F(x) = 31.50 + 32x.

A linear function can be used to represent the total retail cost of renting a vacuum cleaner. Let's denote the fixed fee as $31.50 and the hourly rate as $32. The linear function F for the total retail cost of a vacuum cleaner can be written as:

F(x) = 31.50 + 32x

Where x represents the number of hours the vacuum cleaner is rented for and F(x) gives the total retail cost.

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The flat fee that the store charges is $14 and the cost for 7 hours is $56

A linear equation is on the form:

y = mx + b

where y, x are variables, m is the rate of change and b is the initial value of y.

let f for the total rental cost of a vacuum cleaner for x hours

Using the points (1, 20) and (3, 32) from the table:

f-f1=(f2-f1)/(x2-x1) (x-x1)

f-20=(32-20)/(3-1) (x-1)

f(x)=6x+14

The flat fee that the store charges is $14

The reasonable domain is 1 ≤ x ≤ 12

The cost for 7 hours is:

f(7) = 6(7) + 14 = 46

here is the complete question-

A hardware store rents vacuum cleaners that customers may use for part or all of a day before returning. The store charges a flat fee plus an hourly rate. Part A Write a linear function f for the totall rental cost of the vacuum cleaner. A. f(x)=6x+14 B. f(x)=3x+14 C, f(x)=3x+22 D. f(x)=6x+24 Part B What is a reasonable domain for the function? A. 14 B. 1 C. 0 D. 20