To find the length of the curve defined by $y=\dfrac{1}{3}(x^2+2)^{3/2}$ over the interval $1\leq x\leq 4$, we must integrate the square root of the sum of 1 and the square of the derivative of our function from 1 to 4.
To begin with, we want to find the derivative of our function. In other words, we need to compute $dy/dx$.
The derivative, with the Chain Rule gives us:
$y' = \dfrac{1}{3} \cdot \dfrac{3}{2} \cdot 2x \cdot (x^2+2)^{1/2}$
Simplifying gives:
$y' = x \cdot (x^2+2)^{1/2}$
Next, we substitute $y'$ into the formula for finding the length of a curve:
$L = \int_{a}^{b}\sqrt{1+(y')^2}dx$
We should note that $a = 1$ and $b = 4$ here. We substitute $y'=x \cdot (x^2+2)^{1/2}$ and obtain:
$L = \int_{1}^{4}\sqrt{1+(x \cdot (x^2+2)^{1/2})^2}dx$
We can now evaluate the integral, where we will square the entire derivative and add 1 as being under the square root.
So, finally, evaluating this integral gives us the length of the curve, which in this case is 24.
Therefore, the length of the curve $y=\dfrac{1}{3}(x^2+2)^{3/2}$ over the interval $1\leq x\leq 4$ is 24.
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Patrician saves $ 1.20 everyday while her sister Nicole saves $ 2.80 daily. The two girls saved $ 150.00 together. How much of this amount was saved by Nicole?
Amount saved by Nicole is $ 105
Solution:
Given that Patrician saves $ 1.20 everyday and Nicole saves $ 2.80 daily
The two girls saved $ 150.00 together
To find: Amount saved by Nicole
Let "x" be the number of days for which Patrician and Nicole saved money
Given that they both saved $ 150.00 together
So we can frame a equation as:
[tex]\text{(amount saved by patrician + amount saved by nicole) } \times x = 150.00[/tex]
[tex](1.20 + 2.80) \times x = 150\\\\4x = 150\\\\x = 37.5[/tex]
Therefore amount saved by nicole:
Nicole has saved $ 2.80 daily for "x" days
[tex]\text{ Amount saved by Nicole } = x \times 2.80\\\\\text{ Amount saved by Nicole } = 37.5 \times 2.80 = 105[/tex]
Thus amount saved by Nicole is $ 105
What is the value of x in the inequality start fraction seven minus two x over negative four end fraction plus two less than negative x ?
A. x less than start fraction one over six end fraction
B. x less than negative start fraction one over six end fraction
C. x greater than negative start fraction one over six end fraction
D. x < 6
Answer:
[tex]x<-\frac{1}{6}[/tex]
Step-by-step explanation:
[tex]\frac{7-2x}{-4} +2<-x[/tex]
Subtract 2 from both sides
[tex]\frac{7-2x}{-4} <-x-2[/tex]
mutliply both sides by -4, flip the inequality
[tex]7-2x >4x+8[/tex]
Add 2x on both sideds
[tex]7>6x+8[/tex]
Subtract 8 from both sides
[tex]-1>6x[/tex]
Divide both sides by 6
[tex]x<-\frac{1}{6}[/tex]
To solve for x, the given inequality (7 - 2x) / (-4) + 2 < -x can be simplified by isolating the variable on one side of the inequality. The value of x is found to be x < -15/2, making option C) x greater than negative one over six the correct answer choice.
Explanation:The given inequality is:
(7 - 2x) / (-4) + 2 < -x
To solve for x, we need to isolate it on one side of the inequality. Let's simplify the equation step by step:
(7 - 2x) / (-4) + 2 < -x
7 - 2x + 8 < -4x (Added 2 to both sides)
15 - 2x < -4x
15 < 2x - 4x (Moved -2x to the right side)
15 < -2x
-15/2 > x
So the value of x in the inequality is x < -15/2. Therefore, the correct answer is option C) x greater than negative start fraction one over six end fraction.
If n and y are positive integers and 450y = n³, which of the following must be an integer?"
I. [tex]\frac{y}{3*2^2*5}[/tex]
II. [tex]\frac{y}{3^2*2*5}[/tex]
III. [tex]\frac{y}{3*2*5^2}[/tex]
(A) None
(B) I only
(C) II only
(D) III only
(E) I, II, and III
Answer:
(B) I only
Step-by-step explanation:
450y = n³
y = n³ / 450 = n³ / (3² * 2 * 5²)
in order to keep y and n be positive integer, the minimal requirement for n³ is n³ = (3³ * 2³ * 5³)
y = n³ / 450
= n³ / (3² * 2 * 5²)
= (3³ * 2³ * 5³) / (3² * 2 * 5²)
= 3*2²*5
∴ I. y / (3*2²*5) = ((3³ * 5³ * 2³) / (3² * 5² * 2)) / (3*2²*5) = 1 ... that keep answer as the smallest positive integer .... Correct answer
II. y / (3²*2*5) = ((3³ * 5³ * 2³) / (3² * 5² * 2)) / (3²*2*5) = 2/3 ...not integer
III. y / (3²*2*5) = ((3³ * 5³ * 2³) / (3² * 5² * 2)) / (3*2*5²) = 2/5 ...not integer
Answer: The correct answer is neither
Step-by-step explanation:
for DeltaMath.
Two new drugs are to be tested using a group of 60 laboratory mice, each tagged with a number for identification purposes. Drug A is to be given to 21 mice, drug B is to be given to another 21 mice, and the remaining 18 mice are to be used as controls. How many ways can the assignment of treatments to mice be made? (A single assignment involves specifying the treatment for each mouse—whether drug A, drug B, or no drug.) (Enter the exact number or an equivalent algebraic expression.)
To find out the number of ways the mice can be assigned to receive the two different drugs or no drug, we use the combination formula. The total number of ways is given by the product of the combinations C(60, 21) * C(39, 21) * 1.
Explanation:In this problem, we are given a group of 60 mice and we need to find the number of ways the mice can be assigned to receive drug A, drug B, or no drug. This is a problem of combinatorics, specifically a problem of combinations.
We have 60 mice and we want to choose 21 for the drug A. The number of ways to do this is given by the combination formula C(n, k) = n! / [k!(n-k)!], where 'n' is the total number of items, 'k' is the number of items to choose, and '!' represents the factorial function. So for drug A, it will be C(60, 21).
Then, from the remaining 39 mice, we need to choose 21 for drug B. The number of ways we can do this is C(39, 21).
Finally, the remaining 18 mice will serve as the control group. Given that there's no need to specifically choose which mice are in the control group (since all the remaining ones default to it), we only have 1 way for this selection.
Therefore, the total number of ways the assignment can be made is the product of the number of ways we can form each group, which is C(60, 21) * C(39, 21) * 1.
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The number of ways the assignment of treatments to the mice can be made is 10278857927713471414080000.
To determine how many ways the treatments can be assigned to the 60 laboratory mice (21 for drug A, 21 for drug B, and 18 as controls), we need to calculate the number of permutations of the assignments.
The total number of mice is 60, consisting of:
- 21 mice for drug A
- 21 mice for drug B
- 18 mice as controls
The number of ways to assign treatments is the number of permutations of these groups within the total set of mice. This can be calculated using the multinomial coefficient:
[tex]\[ \frac{60!}{21! \times 21! \times 18!} \][/tex]
Where:
- 60! is the factorial of 60 (total number of mice)
- 21! is the factorial of 21 (number of mice for drug A)
- 21! is again the factorial of 21 (number of mice for drug B)
- 18! is the factorial of 18 (number of control mice)
Let's compute this step-by-step:
1. Calculate 60!:
[tex]\[ 60! = 60 \times 59 \times 58 \times \ldots \times 2 \times 1 \][/tex]
2. Calculate 21!:
[tex]\[ 21! = 21 \times 20 \times \ldots \times 2 \times 1 \][/tex]
3. Calculate 18!:
[tex]\[ 18! = 18 \times 17 \times \ldots \times 2 \times 1 \][/tex]
Now, plug these into the formula:
[tex]\[ \frac{60!}{21! \times 21! \times 18!} = \frac{60 \times 59 \times \ldots \times 2 \times 1}{(21 \times 20 \times \ldots \times 2 \times 1) \times (21 \times 20 \times \ldots \times 2 \times 1) \times (18 \times 17 \times \ldots \times 2 \times 1)} \][/tex]
[tex]\[ \frac{60!}{21! \times 21! \times 18!} = \frac{83209871127413901442763411832233643807541726063612459524492776964096000000000000000}{51090942171709440000} \][/tex]
[tex]\[ \frac{60!}{21! \times 21! \times 18!} = 10278857927713471414080000 \][/tex]
A container is 1/10 full of grain. If 21 additional gallons of grain are added, the container is 4/5 full. What is the volume of the container, in gallons?
Answer:
30 gallons
Step-by-step explanation:
let x represent the full container, then
[tex]\frac{1}{10}[/tex] x + 21 = [tex]\frac{4}{5}[/tex] x
Multiply through by 10 to clear the fractions
x + 210 = 8x ( subtract x from both sides )
210 = 7x ( divide both sides by 7 )
30 = x
The volume of the container is 30 gallons
Answer: the volume of the container is 30 gallons
Step-by-step explanation:
Let V represent the volume of the container, in gallons.
The initial volume of grains in the container is 1/0 × V = V/10
If 21 additional gallons of grain are added, the container is 4/5 full. This means that 21 gallons of grain + V/10 will occupy 4/5 of the volume of the container. Therefore
4/5 × V = 21 + V/10
4V/5 = 21 + V/10
4V/5 - V/10 = 21
7V/10 = 21
Multiplying the left hand side and right hand side of the equation by 10, it becomes
7V/10 × 5 = 21 × 10 = 210
7V = 210
V = 210/7 = 30
An aerial photograph from a U-2 spy plane is taken of a building suspected of housing nuclear warheads. When the photograph is taken, the angle of elevation of the sun is 30∘. By comparing the shadow cast by the building in question to the shadows of other objects of known size in the photograph, scientists determine that the shadow of the building in question is 98 feet long.How tall is the bulding? (Round your answer to two decimal places.)
Answer:
Building is 56.58 feet long.
Step-by-step explanation:
Consider the provided information.
An aerial photograph from a U-2 spy plane is taken of a building suspected of housing nuclear warheads.
When the photograph is taken, the angle of elevation of the sun is 30°.
By comparing the shadow cast by the building in question to the shadows of other objects of known size in the photograph, scientists determine that the shadow of the building in question is 98 feet long.
As we know: [tex]\tan\theta=\frac{opp}{adj}[/tex]
The value of tan 30 degrees is [tex]\frac{1}{\sqrt{3} }[/tex]
[tex]\tan30=\frac{h}{98}[/tex]
[tex]\frac{1}{\sqrt{3}}=\frac{h}{98}[/tex]
[tex]h=\frac{98}{\sqrt{3}}[/tex]
[tex]h=56.58[/tex]
Hence, Building is 56.58 feet long.
Answer:
56.58 feet long
Step-by-step explanation:
Consider the provided information.
An aerial photograph from a U-2 spy plane is taken of a building suspected of housing nuclear warheads.
When the photograph is taken, the angle of elevation of the sun is 30°.
By comparing the shadow cast by the building in question to the shadows of other objects of known size in the photograph, scientists determine that the shadow of the building in question is 98 feet long.
As we know:
The value of tan 30 degrees is
Hence, Building is 56.58 feet long.
Given the function ƒ(x) = 8(x - 4) - 18, determine the value of x such that ƒ(x) = 22. Select one: A. 3 B. 6 C. 9 D. 12
Answer:
C. 9
Step-by-step explanation:
it's multiple choice so plug each value in
A = -26
B = -34
C = 22
D = 46
so the answer is c
A square is inscribed in a right triangle so that they have a common right angle. The legs of the triangle are 6 in and 8 in long. Find the length of the side of the square.
Answer:
24/7 = 3 3/7 inches
Step-by-step explanation:
The triangle can be represented by a line in the first quadrant with y-intercept 8 and x-intercept 6. Then its equation is ...
x/6 +y/8 = 1 . . . . . intercept form of the equation of the line
4x + 3y = 24 . . . . multiply by 24 to get standard form
Since the square will have the origin as one corner, and all sides are the same length, the opposite corner will lie on the line y=x. Then we're solving the system ...
4x +3y = 24
y = x
to find the side length.
__
By substitution for y, this becomes ...
4x +3x = 24
7x = 24
x = 24/7 = 3 3/7
The length of the side of the square is 3 3/7 inches.
The question involved finding the side length of a square inscribed in a right triangle. The correct approach uses additional geometry to set up a system of equations resulting from segments on the legs of the triangle equal to the side length of the square. Solving these equations reveals that the side length of the square is 2 inches.
Explanation:The student is seeking to find the length of the side of a square inscribed in a right triangle with legs measuring 6 inches and 8 inches. To determine this, one can use the Pythagorean theorem which relates the legs of a right triangle to its hypotenuse. However, in this scenario, the side of the square also acts as a 'leg' of two smaller right triangles within the original triangle. The length of the square, let's call it 's', plus the square's length (again 's') will equal the longer leg of the triangle (8 inches); similarly, 's' plus the length from the corner of the square to the right angle of the triangle (which is also 's') will equal the shorter leg (6 inches).
Therefore, we have two equations: 2s = 8 and 2s = 6. Since both cannot be true with the same value of 's', we realize that the premise of the question must be reconsidered. The actual process involves a bit more geometry, using the fact that the segments along the legs that are not part of the square must be equal respectively, leading to a system of equations to solve for the length of the side of the square. Let's denote these segments as 'x'; hence the equations are s + x = 6 and s + x = 8. Subtracting these equations from the original leg lengths gives us x = 6 - s and x = 8 - s. As these segments are equal, we can set them equal to each other, getting 6 - s = 8 - s, which simplifies to s = 2 inches.
6. If X is the midpoint of WY, WX = 3x - 1 and
WY = 10x - 26, find XY.
XY = 17
Step-by-step explanation:
It is given that x is the mid-point of WY, so it divides WY in two equal parts
WX and XY so the sum of both lengths will constitute the length of WY
[tex]WY = WX+XY\\XY = WY - WX[/tex]
putting values
[tex]XY = 10x-26 -(3x-1)\\= 10x-26-3x+1\\= 7x-25[/tex]
As X is the mid-point,
[tex]WX = XY\\3x-1 = 7x-25\\3x-1-3x = 7x-25-3x\\-1 = 4x-25\\-1+25 = 4x-25+25\\24 = 4x[/tex]
Dividing both sides by 4
[tex]\frac{4x}{4} = \frac{24}{4}\\x = 6[/tex]
XY = 7x-25
[tex]=7(6) -25\\= 42-25\\=17[/tex]
So,
XY = 17
Verification:
WX = XY
[tex]3x-1 = 7x-25\\3(6)-1 = 7(6)-25\\18-1 = 42-25\\17 = 17[/tex]
Keywords: Linear equations, polynomials
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One end point of a line segment is(-3,-6). The length of the line segment is 7 units. Find four points that could serve as the other end point of the given line segment.
Answer:
(-10, -6), (4, -6), (-3, -13), (-3, 1)
Step-by-step explanation:
The easiest points to find that have rational coordinates are the ones 7 units up or down, left or right from the given point. Those are listed above.
We used the distance formula to find four points that could serve as the other end of a line segment with one end at (-3, -6) and length of 7 units. The four points are (0, -6 + √40), (0, -6 - √40), (-3 + √13, 0), and (-3 - √13, 0).
Finding the Other End Point of a Line Segment
To find four points that can serve as the other end point of a line segment with one end point at (-3, -6) and a length of 7 units, we use the distance formula. The distance formula is:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Given, (x₁, y₁) is (-3, -6) and the distance is 7, we need to find (x₂, y₂) such that:
√((x₂ + 3)² + (y₂ + 6)²) = 7
Squaring both sides: (x₂ + 3)² + (y₂ + 6)² = 49
Case 1: Let x₂ = 0(0 + 3)² + (y₂ + 6)² = 49
9 + (y₂ + 6)² = 49
(y₂ + 6)² = 40
y₂ + 6 = ±√40
y₂ = -6 ± √40
(x₂ + 3)² + (0 + 6)² = 49
(x₂ + 3)² + 36 = 49
(x₂ + 3)² = 13
x₂ + 3 = ±√13
x₂ = -3 ± √13
Thus, the four possible points for the other end point of the line segment are (0, -6 + √40), (0, -6 - √40), (-3 + √13, 0), and (-3 - √13, 0).
What is the tens digit of the positive integer r ? 1) The tens digit of r/10 is 3. 2) The hundreds digit of 10r is 6.
Answer:6
Step-by-step explanation:
Let digit be r=abc
if tens digit of [tex]\frac{r}{10}[/tex] is 3.2
i.e. [tex]\frac{r}{10}[/tex] is written as ab.c
so tens digit is a=3
If the hundreds digit of 10r is 6
i.e. 10r is written as abc0
its hundreds digit is b=6
thus tens digit of abc is b=6
To find the tens digit of the integer r, we use the information that the hundreds digit of 10r is 6, indicating that the tens digit of r is also 6.
Explanation:The question is asking to determine the tens digit of a positive integer r. To solve this, we need to analyze the given facts separately:
The tens digit of r/10 is 3. If we divide the integer r by 10 and find that its tens digit is 3, this implies that in the original number r, the ones digit was 3.The hundreds digit of 10r is 6. By multiplying r by 10, we essentially shift each digit one place to the left, meaning the tens digit of r becomes the hundreds digit of 10r. Therefore, if the hundreds digit of 10r is 6, the tens digit of r is also 6.From statement 2 alone, we can determine that the tens digit of the positive integer r is 6.
Today robbie is carrying his history textbook and his lunch in his backpack. If the history textbook weighs 2 5/6 pounds and his lunch weighs 1 2/3 pounds, how much weight is in robbies backpack?
Answer:
37/9 pounds
Step-by-step explanation:
+ them together
Answer:
Step-by-step explanation:
robbie is carrying his history textbook and his lunch in his backpack today.
The history textbook weighs 2 5/6 pounds. Converting 2 5/6 pounds to improper fraction, it becomes
17/6 pounds.
His lunch weighs 1 2/3 pounds. Converting 1 2/3 pounds to improper fraction, it becomes
5/3 pounds.
Total weight of the back backpack today would be the sum of the weight of his history textbook and his lunch. It becomes
17/6 + 5/3 = 27/6 = 4 1/2 pounds
PLEASE HELP ME
Using homeowners insurance of $1012 per year, property taxes 1.1% per year, and the annual 1% of the home's purchase price for maintenance, what is the true cost of a $96,000 home with a 30 year mortgage payment of $545.08 per month?
A. $301,012.43
B. $225,907.80
C. $287,068.80
D. $299,007.45
When considering mortgage payments, homeowners insurance, property tax and maintenance costs over 30 years, the true cost of the $96,000 home is $287,068.80.
Explanation:The true cost of a home not only includes the original purchase price but also takes into account annual homeowners insurance, property taxes, and maintenance costs. In this case, the purchase price is $96,000, homeowners insurance costs $1012 per year, property taxes are 1.1% of the home price each year, and annual home maintenance is 1% of the home price.
First, calculate the total mortgage payments made over 30 years. The monthly payment is $545.08, so the total mortgage payments are:
30 years * 12 months/year * $545.08/month = $196,228.80
Next, calculate the total cost of homeowners insurance over 30 years:
30 years * $1012/year = $30,360
Then, calculate the total property taxes over 30 years:
30 years * 1.1% of $96,000/year = $31,680
Lastly, calculate the total home maintenance over 30 years:
30 years * 1% of $96,000/year = $28,800
Adding all of these costs together gives the total cost of the home:
$196,228.80 + $30,360 + $31,680 + $28,800 = $287,068.80
So, the correct answer is option C: $287,068.80.
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Only questions 10-12!! Help please!!
Answer:
Step-by-step explanation:
10) The opposite sides of a parallelogram are equal. It means that
a + 15 = 3a + 11
3a - a = 15 - 11
2a = 4
a = 4/2 = 2
Also,
3b + 5 = b + 11
3b - b = 11 - 5
2b = 4
b = 4/2 = 2
11) The opposite angles of a parallelogram are congruent and the adjacent angles are supplementary. This means that
2x + 11 + x - 5 = 180
3x + 6 = 180
3x = 180 - 6 = 174
x = 174/3 = 58
Therefore,
2x + 11 = 2×58 + 11 = 127 degrees
The opposite angles of a parallelogram are congruent, therefore,
2y = 127
y = 127/2 = 63.5
12) The diagonals of a parallelogram bisect each other. This means that each diagonal is divided equally at the midpoint. Therefore
3y - 5 = y + 5
3y - y = 5 + 5
2y = 10
y = 10/2 = 5
Also,
z + 9 = 2z + 7
2z - z = 9 - 7
z = 2
What is the measure of an interior angle in a regular triangle? Write your answer as an integer or as a decimal rounded to the nearest tenth.
Answer:
The measure of an interior angle in a regular triangle is 60 degrees
Step-by-step explanation:
we know that
A regular triangle has three equal sides and three equal interior angles
Remember that the sum of the interior angles in a triangle must be equal to 180 degrees
so
Divide 180 by 3 to determine the measure of each interior angle
[tex]\frac{180^o}{3}=60^o[/tex]
A regular triangle is called equilateral triangle
therefore
The measure of an interior angle in a regular triangle is 60 degrees
tyler is 4 years older than tiffany in ten years he will be twicse as old a tifinay.
For this case we have to:
x: Variable that represents Tyler's age
y: Variable that represents Tiffany's age
According to the data of the statement we can propose the following equations:
[tex]x = y + 4\\x + 10 = 2y[/tex]
To solve we substitute the first equation in the second equation:
[tex]y + 4 + 10 = 2y\\14 = 2y-y\\14 = y[/tex]
So, Tiffany is 14 years old.
Then Tyler has:
[tex]x = 14 + 4\\x = 18[/tex]
Tyler is 18 years old.
Answer:
Tyler: 18 years old
Tiffany: 14 years old
Lindsay can paint 1/x of a certain room in 20 minutes. What fraction of the same room can Joseph paint in 20 minutes if the two of them can paint the room in an hour, working together at their respective rates?
a) 1/3x
b) 3x/x-3
c) x-3/3x
d) x/x-3
e) x-3/x
Answer:
[tex]\frac{x-3}{3x}[/tex]
Step-by-step explanation:
Lindsay can paint 1/x of a certain room in 20 minutes.
1 hour = 3 times 20 minutes
rate of work by Lindsay in 20 minutes is [tex]\frac{3}{x}[/tex]
Let 't' be the work done by Joseph
rate of work by Joseph in 20 minutes is [tex]\frac{3}{t}[/tex]
Both completed the work in 1 hour
[tex]\frac{3}{x} +\frac{3}{t} =1[/tex]
solve the equation for 't'
Subtract 3/x on both sides
[tex]\frac{3}{t} =1-\frac{3}{x}[/tex]
[tex]\frac{3}{t} =\frac{x-3}{x}[/tex]
cross multiply it
[tex]3x=t(x-3)[/tex]
Divide both sides by x-3
[tex]\frac{3x}{x-3} =t[/tex]
Work done together is
[tex]\frac{x-3}{3x}[/tex]
State the horizontal asymptote is the rational function. F(x)=x+9/x^2+8x+8
None
Y=x
Y=9
Y=0
Answer:
y = 0
Step-by-step explanation:
When the degree of the denominator of a rational function is greater than the degree of the numerator, then the equation of the horizontal asymptote is
y = 0
here degree of numerator is 1 and degree of denominator is 2
Degree of denominator > degree of numerator, thus
y = 0 ← equation of horizontal asymptote
Please Help!! I'm soo confused.
A line contains the points (-30, 42) and (-27, -30) what is the slope of the line in simplest form?
if you need any other information let me know!!
:))
We can use the points (-30, 42) and (-27, -30) to solve.
Slope formula: y2-y1/x2-x1
= -30-42/-27-(-30)
= -72/3
= -24 (Answer)
Best of Luck!
Find the area of a rhombus with side length 6 and an interior angle with measure $120^\circ$.
Answer:
The area of a rhombus is [tex]18\sqrt{3}[/tex] square units.
Step-by-step explanation:
Side length of rhombus = 6 units.
Interior angle of rhombus = 120°
another Interior angle of rhombus = 180°-120° = 60°.
Draw an altitude.
In a right angled triangle
[tex]\sin \theta=\frac{opposite}{hypotenuse}[/tex]
[tex]\sin (60)=\frac{h}{6}[/tex]
[tex]\frac{\sqrt{3}}{2}=\frac{h}{6}[/tex]
Multiply both sides by 6.
[tex]3\sqrt{3}=h[/tex]
The height of the rhombus is [tex]3\sqrt{3}[/tex].
Area of a rhombus is
[tex]Area=base\times height[/tex]
[tex]Area=6\times 3\sqrt{3}[/tex]
[tex]Area=18\sqrt{3}[/tex]
Therefore, the area of a rhombus is [tex]18\sqrt{3}[/tex] square units.
Area of Rhombus is 31.176 unit² (Approx.)
Given that;Length of rhombus side = 6 unit
Angle = 120°
Find:Area of Rhombus
Computation:Area of Rhombus = Side²(Sin θ)
Area of Rhombus = 6²(Sin 120)
Area of Rhombus = 36(0.866)
Area of Rhombus = 31.176 unit² (Approx.)
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A ribbon 56 cm long is cut into two pieces. One of the pieces is three times longer than the other. Find the length, in centimeters, of both pieces of ribbons
Answer:one piece is 42 centimeters and the other piece is 14 centimeters
Step-by-step explanation:
Let x represent the length, in centimeters, of one piece of the ribbon.
Let y represent the length, in centimeters, of the other piece of the ribbon.
The ribbon, 56 cm long is cut into two pieces. This means that
x + y = 56 - - - - -- - - - -1
One of the pieces is three times longer than the other. This means that
x = 3y
Substituting x = 3y into equation 1, it becomes
3y + y = 56
4y = 56
y = 56/4 = 14 centimeters
x = 3y = 2×14 = 42 centimeters
How many triangles and quadrilaterals altogether can be formed using the vertices of a 7-sided regular polygon?
A. 35
B. 40
C. 50
D. 65
E. 70
Answer: The correct option is
(E) 70.
Step-by-step explanation: We are given to find the number of triangles and quadrilaterals altogether that can be formed using the vertices of a 7-sided regular polygon.
To form a triangle, we need any 3 vertices of the 7-sided regular polygon. So, the number of triangles that can be formed is
[tex]n_t=^7C_3=\dfrac{7!}{3!(7-3)!}=\dfrac{7\times6\times5\times4!}{3\times2\times1\times4!}=35.[/tex]
Also, to form a quadrilateral, we need any 4 vertices of the 7-sided regular polygon. So, the number of quadrilateral that can be formed is
[tex]n_q=^7C_4=\dfrac{7!}{4!(7-4)!}=\dfrac{7\times6\times5\times4!}{4!\times3\times2\times1}=35.[/tex]
Therefore, the total number of triangles and quadrilaterals is
[tex]n=n_t+n_q=35+35=70.[/tex]
Thus, option (E) is CORRECT.
Only questions 8 and 9 help please!!!
Answer:
Step-by-step explanation:
Perpendicular slopes are the opposite reciprocals of the slopes given. Our slope in 8 is -2. That means that the perpendicular slope is 1/2. If the line goes through (2, -1), then
[tex]y-(-1)=\frac{1}{2}(x-2)[/tex] and
[tex]y+1=\frac{1}{2}x-1[/tex] and
[tex]y=\frac{1}{2}x-2[/tex]
9 is a tiny bit trickier because we don't have the slope, the x term, on the opposite side of the equals sign from the y. Let's do that and then we can determine the slope of that given line. Moving over the 3x and isolating the y:
y = -3x + 5
So the slope is -3. That means that the perpendicular slope is 1/3. If the line goes through (-9, 3), then
[tex]y-3=\frac{1}{3}(x-(-9))[/tex] and
[tex]y-3=\frac{1}{3}(x+9)[/tex] and
[tex]y-3=\frac{1}{3}x+3[/tex] so
[tex]y=\frac{1}{3}x+6[/tex]
Answer:
Step-by-step explanation:
8) y = -2x + 1 and the line passes through (2, - 1)
The equation of a straight line can be represented in the slope intercept form as
y = mx + c
Where
m = slope = (change in the value of y in the y axis) / (change in the value of x in the x axis)
The equation of the given line is
y= -2x + 1
Comparing with the slope intercept form, slope = - 2
If two line passing through the given point is perpendicular to the given line, it means its slope is the negative reciprocal of that of the given line. Therefore, the slope of the line passing through (2,-1) is 1/2
To determine the intercept, we would substitute m = 1/2, x = 2 and y = -1 into y = mx + c. It becomes
- 1 = 1/2 × 2 + c = 1 + c
c = - 1 - 1 = - 2
The equation becomes
y = x/2 - 2
9) 3x + y = 5 and the line passes through (-9, 3)
The equation of the given line is
3x + y = 5
y = -3x + 5
Comparing with the slope intercept form, slope = - 3
If two line passing through the given point is perpendicular to the given line, it means its slope is the negative reciprocal of that of the given line. Therefore, the slope of the line passing through (- 9, 3) is 1/3
To determine the intercept, we would substitute m = 1/3, x = -9 and y = 3 into y = mx + c. It becomes
3 = 1/3 × -9 + c = - 3 + c
c = 3 + 3 = 6
The equation becomes
y = x/3 + 6
An 85-foot rope from the top of a tree house to the ground forms a 45 degree angle of elevation from the ground. How high is the top of the tree house?
Answer:
(about) 61.1
Step-by-step explanation:
To find the height, or H, we can use
sin (45 degree angle) =H/85
rearrange as H=85* sin (45 degrees) =(about) 61.1
Sayuri's Asian Café makes the best pot stickers in town. The kitchen's production is usually between 20 and 22 pot stickers per hour. Sayuri buys a new machine to help the team make pot stickers faster. She tracks production over the course of seven days. On which day does the machine make a positive impact on production?
The question is not complete without a picture depicting the pot stickers production. I found a similar question with a table for the production of pot stickers which is attached to this answer. If the actual table is different from the table in this answer, you can still answer your question accordingly using my working and reasoning.
Answer:
Day 6
Step-by-step explanation:
It was given that the usual production of the kitchen is between 20 to 22 pot stickers per hour. Day 1 to 3, the production is within the range given.
However on day 4 and 5, the production is dropped to 10 and 15 - probably due to the workers unfamiliar with the machine.
On day 6 and 7, the production increases to 50 and 55. Therefore the day the machine makes positive impact is on day 6 where the production starts to make a significant increase.
We cannot determine the exact day when the new machine makes a positive impact on production at Sayuri's Asian Café.
Explanation:The question asks on which day the new machine makes a positive impact on production at Sayuri's Asian Café. To determine this, we need to compare the production before and after the machine was introduced. Unfortunately, the statement provided does not mention the production levels before the machine. Therefore, we cannot determine the exact day when the machine makes a positive impact.
What is the area of the new trapezoid formed by dilating the original by a factor of 3
Answer:
[tex]A_{new}=9A_{original}[/tex]
Step-by-step explanation:
The are of a trapezoid is:
[tex]A_{original}=\frac{(a+b)}{2}h[/tex]
where:
a and b are basesh is the heightWhen a geometric figure dilates, every coordinate of the original figure must be multiplied by the scale factor of this dilatation. In our case this factor is 3, therefore we will have:
[tex]A_{new}=\frac{(3a+3b)}{2}3h[/tex]
[tex]A_{new}=9\frac{(a+b)}{2}h[/tex]
[tex]A_{new}=9A_{original}[/tex]
The new area is 9 times the original one.
I hope it helps you!
Explain how to find n, the number of copies the machine can print in one minute. I need an algebraic expression for the answer.
Once the first part is done, I need help with this question.
Working at the same rate, how long will it take the machine to print 5,200 copies? Explain how you found your answer.
Answer:
It prints 65 copies in 1 minute.
It takes 80 minutes to print 5200 copies.
Step-by-step explanation:
In 5 minutes the machine prints 325 copies and it can print at that steady state.
We have to find the number of copies it prints in 1 minute.
So ,number of copies in 1 minute = [tex]\frac{number of copies in 5 minute}{5}[/tex]
= [tex]\frac{325}{5}[/tex]
= 65
Hence it prints 65 copies in 1 minute.
The time taken to print 5200 copies = [tex]time to make 1 copy \times 5200[/tex]
= [tex]\frac{1}{65} \times 5200[/tex]
= 80 minutes
Charlotte pays $24 for 3 yards of landscaping fabric and 6 yards of chicken wire. Kami pays $30 for 6 yards of landscaping fabric and 3 yards of chicken wire. What is the price of 1 yard of chicken wire?
Answer: the price of 1 yard of chicken wire is $2
Step-by-step explanation:
Let x represent the cost of one yard of landscaping fabric.
Let y represent the cost of one yard of chicken wire.
Charlotte pays $24 for 3 yards of landscaping fabric and 6 yards of chicken wire. This means that
3x + 6y = 24 - - - - - - - - - - 1
Kami pays $30 for 6 yards of landscaping fabric and 3 yards of chicken wire. This means that
6x + 3y = 30 - - - - - - - - -2
Multiplying equation 1 by 6 and equation 2 by 3, it becomes
18x + 36y = 144
18x + 9y = 90
Subtracting
27y = 54
y = 54/27 = 2
Substituting y = 2 into equation 1, it becomes
3x + 6×2 = 24
3x = 24 - 12 = 12
x = 12/3 = 4
Answer:
$2 per yard.
Step-by-step explanation:
Let the price per yard of landscaping fabric be f and the price of chicken wire per yard be c.
3f + 6c = 24 Multiply this equation by - 2:
-6f - 12c = -48 Also:
6f + 3c = 30
Adding the last 2 equations:
-9c = -18
c = 2.
In Fig .1 2-40,one end of a uniform beam of weight 222N is hinged to a wall ;the other end is sup -ported by a wire that makes angles 6 =30.0with both wall and beam .Find (a)thetension in the wire and the ( b)horizontaland (c)verticalcompo -nents of the force of the hinge on the beam .
Answer:
(a) 74√3 N
(b) 37√3 N
(c) 111 N
Step-by-step explanation:
(a) The moment about the hinge produced by the beam is the product of the weight of the beam and the distance of its center from the wall. The tension in the wire counteracts that moment.
The tension in the wire acts at a horizontal distance from the wall that is twice the distance to the beam's center, so the tension's vertical component is only half the weight of the beam.
Since the wire is at a 30° angle to the wall, the horizontal component of the tension is 1/√3 times the vertical component. Altogether, the tension in the wire is (2/√3) times half the beam's weight, or 74√3 N.
__
(b) The horizontal force at the hinge counteracts the horizontal component of the tension in the wire, so is 111/√3 N = 37√3 N.
__
(c) The vertical component of the force at the hinge is half the beam weight, so is 111 N.
a) The tension in the wire is 192.3 N. b) The horizontal component of the force of the hi-nge on the beam is 96.1 N. c) Vertical component of the force of the hi-nge on the beam is 55.5 N.
(a) To find the tension in the wire, we can use a torque balance around the hi-nge. The torques due to the weight of the beam and the tension in the wire are equal and opposite, so we have:
T * L * cos(30°) = W * L/2
where:
T is the tension in the wire
L is the length of the beam
W is the weight of the beam
Solving for T, we get:
T = W * cos(30°) / 2
Substituting the known values, we get:
T = 222 N * cos(30°) / 2
= 192.3 N
Therefore, the tension in the wire is 192.3 N.
(b) The horizontal component of the force of the hi-nge on the beam is equal to the tension in the wire multiplied by the sine of the angle between the wire and the beam. This angle is 30°, so we have:
[tex]F_h[/tex] = T * sin(30°)
Substituting the known values, we get:
[tex]F_h[/tex] = 192.3 N * sin(30°)
= 96.1 N
Therefore, the horizontal component of the force of the hi-nge on the beam is 96.1 N.
(c) The vertical component of the force of the hi-nge on the beam is equal to the weight of the beam minus the tension in the wire multiplied by the cosine of the angle between the wire and the beam. This angle is 30°, so we have:
[tex]F_v[/tex] = W - T * cos(30°)
Substituting the known values, we get:
[tex]F_v[/tex] = 222 N - 192.3 N * cos(30°)
= 55.5 N
Therefore, the vertical component of the force of the hi-nge on the beam is 55.5 N.
To learn more about horizontal component here:
https://brainly.com/question/32317959
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Andrew's bowling scores are approximately normally distributed with mean 130 and standard deviation 21, while Pam's scores are normally distributed with mean 125 and standard deviation 12. If Andrew and Pam each bowl one game, then assuming that their scores are independent random variables, approximate the probability that the total of their scores is above 265
Answer:
2
Step-by-step explanation: