A triangle has angle measures of (x + 3)°, (5x – 8)°, and (2x + 1)°.

What is the measure of the smallest angle of the triangle in degrees?

A 47°

B 26°

C 107°

D 23°

The total cost of producing 100 bar stools using the left-rectangle approximation with five rectangles is approximately $200.

The marginal cost function for producing the x-th item of a certain brand of bar stool is given by MC(x) = 20 - 0.5x, where 0 ≤ x ≤ 100. To estimate the total cost of producing 100 bar stools using the left-rectangle approximation with five rectangles, we first need to calculate the width of each rectangle.

The width of each rectangle is (100 - 0) / 5 = 20.

Now, we will evaluate the marginal cost function at the left endpoints of each rectangle:
MC(0) = 20 - 0.5(0) = 20
MC(20) = 20 - 0.5(20) = 10
MC(40) = 20 - 0.5(40) = 0
MC(60) = 20 - 0.5(60) = -10
MC(80) = 20 - 0.5(80) = -20

Next, multiply each value by the width (20) and sum the results to approximate the variable cost:
Variable cost ≈ 20(20) + 10(20) + 0(20) + (-10)(20) + (-20)(20) = 400 + 200 - 200 - 400 = 0

Finally, add the fixed cost of $200 to the variable cost:
Total cost ≈ $200 + $0 = $200

So, the total cost of producing 100 bar stools using the left-rectangle approximation with five rectangles is approximately $200.

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To find the relative frequency of boys who want to go to the water park, we need to divide the number of boys who want to go to the water park by the total number of boys in the sample:

Relative frequency = Number of boys who want to go to the water park / Total number of boys

Assuming you have the necessary data, you can compute this value by dividing the number of boys who want to go to the water park by the total number of boys and rounding the result to the nearest hundredth.

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The value of M is 4.

The value of M represents the length of the segment that connects the midpoints of the non-parallel sides of trapezoid ABCD. To find the value of M, we can use the fact that the trapezoids are similar.

Now, let's label the points where MN intersects sides BC and FG as P and Q, respectively. We can use the fact that MN is parallel to sides BC and FG to show that triangles BMP and FQN are similar to trapezoids ABCD and EFGH, respectively.

Therefore, we can write:

BM/AB = MP/CD = k

and

FQ/EF = QN/GH = k

Since we know that AB/EF = k, we can substitute this into the first equation to get:

BM/EF = MP/GH

Similarly, we can substitute FQ/EF = k into the second equation to get:

FQ/EF = QN/GH

Now, let's combine these two equations to get:

(BM + FQ)/EF = (MP + QN)/GH

But we know that BM + FQ = EF, and MP + QN = GH. Therefore, we can simplify the equation to get:

EF/EF = GH/GH

Or simply:

1 = 1

This means that our calculations are correct, and that we can use the ratio k to find the value of M. Specifically, since MN is parallel to AB and CD, we know that the length of MN is equal to half the difference between the lengths of AB and CD. Therefore, we can write:

M = (1/2)(AB - CD)/k

When we apply the values on it, then we get,

M = (1/2) (56 - 48) / 1

M = (1/2) x 8 = 4

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To generate the sequence, we start with a1 = 4, and then use the formula an = an-1 + 7 for n > 1.

So, to find the first 3 terms of the sequence, we can use the formula:

a2 = a1 + 7 = 4 + 7 = 11

a3 = a2 + 7 = 11 + 7 = 18

The first 3 terms of the sequence are 4, 11, and 18.

So, the answer is 4, 11, 18, which corresponds to the third option.

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Answer:

3x + 5 = 4x - 40

x = 45

5y - 50 = y

4y = 50

y = 25/2

Answer:

y = 9

Step-by-step explanation:

• Parallel lines have equal slopes

calculate the slope m of  TU using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = T(- 7, 4 ) and (x₂, y₂ ) = U(- 2, - 6 )

[tex]m_{TU}[/tex] = [tex]\frac{-6-4}{-2-(-7)}[/tex] = [tex]\frac{-10}{-2+7}[/tex] = [tex]\frac{-10}{5}[/tex] = - 2

now calculate the slope of VW and equate to - 2

with (x₁, y₁ ) = V(8, 7 ) and (x₂, y₂ ) = W(7, y )

[tex]m_{VW}[/tex] = [tex]\frac{y-7}{7-8}[/tex] = [tex]\frac{y-7}{-1}[/tex]

now equating gives

[tex]\frac{y-7}{-1}[/tex] = - 2 ( multiply both sides by - 1 to clear the fraction )

y - 7 = 2 ( add 7 to both sides )

y = 9

In ΔMNO, possible values of ∠N are 64.6° and 115.4°.

To find the possible values of ∠N, follow these steps:

1. Since the sum of angles in a triangle is 180°, we first find ∠O by subtracting ∠M from 180°: 180° - 38° = 142°.

2. Next, we use the Law of Sines to find the sine of ∠N: sin(∠N) = (n * sin(∠O)) / m = (88 * sin(142°)) / 60.

3. Solve for sin(∠N), which gives us two possible values: sin(∠N) ≈ 0.8988 and sin(∠N) ≈ -0.8988.

4. Find the inverse sine (arcsin) of both values to get the possible angles for ∠N: arcsin(0.8988) ≈ 64.6° and arcsin(-0.8988) ≈ 115.4° (adding 180° to the negative result).

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Answer:

Problem 8:
Option C)       DE = √21

Problem 9:
Option C)     Perimeter = 30 cm

Step-by-step explanation:

1. Problem 8

Draw a line segment connecting the center of the circle, X, to A. This is the radius of the circle

The points ABX form a right triangle with AX as the hypotenuse and AB, BX as the legs of the right triangle

By the Pythagorean theorem
hypotenuse² = sum of the squares of the two legs

Plugging in line segment references

AX² = AB² + BX²

We are given AC = 8, BX = 3

Since the segment BX intersects AC at right angles, AB = BC = AC/2

So AB = 8/2= 4

Plugging these values into the Pythagorean formula
AX² = AB² + BX²

AX² = 4² +3²

AX² = 16 + 9

AX² = 25

Draw a line connecting X and D
The points DEX form a right triangle with DX as the hypotenuse and EX and DE as the legs

Again, by the Pythagorean theorem

DX² = DE² + EX²

But DX is the radius of the circle so it must be the same as AX

Substitute for DX in terms of AX
DX² = DE² + EX²

=> AX² = DE² + EX²

But AX² = 25 and EX = 2 giving EX² = 4

Therefore

AX² = DE² + EX² becomes
25 = DE² + 4

DE² = 25 - 4

DE² = 21

DE = √21

This is choice C

Problem 9

To find the perimeter, just add up the individual side lengths:

Perimeter = 10 + 8 + 7 + 5 = 30 cm

Option C

Answer:

p=3

Step-by-step explanation:

Answer:

2.2

Step-by-step explanation:

To find your GPA, use the attached image:

After you've written down your numerical scores, divide it by the total classes you're taking, which is 7.

So, your GPA is 2.2

To determine if two lines intersect, compare their slopes and y-intercepts, and solve their equations simultaneously.

To determine if two lines intersect, you need to compare their key features, such as their slope and y-intercept.

If the slopes of the two lines are different, then they will intersect at some point.

If the slopes are the same, then the lines may or may not intersect, depending on whether or not their y-intercepts are also the same.

To find the point of intersection, you can set the two linear functions equal to each other and solve for the variable. The resulting value will give you the x-coordinate of the intersection point, which can then be substituted back into either equation to find the corresponding y-coordinate.

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a. There would be  at least 20 students must go on the trip for the average cost per student to fall to $175.

b. As more students go on the trip, the total cost of the trip increases, but the cost per student decreases.

a. To find out how many students must go on the trip for the average cost per student to fall to $175, we can use the formula:

Total Cost = $500 + $150 x Number of Students

Average Cost per Student = Total Cost / Number of Students

Setting the average cost per student to $175 and solving for the number of students, we get:

$175 = ($500 + $150 x Number of Students) / Number of Students

Multiplying both sides by Number of Students, we get:

$175 x Number of Students = $500 + $150 x Number of Students

Simplifying, we get:

$25 x Number of Students = $500

Number of Students = $500 / $25 = 20

Therefore, at least 20 students must go on the trip for the average cost per student to fall to $175.

b. As more students go on the trip, the total cost of the trip increases, but the cost per student decreases. This is because the fixed cost of $500 is spread over more students, making the cost per student lower. So, as more students go on the trip, the average cost per student will continue to decrease.

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The point where line A intersects line B is [2, 10].

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁)

Where:

At data point (1, 10 1/3) and a slope of -1/3, a linear equation for this line can be calculated or determined by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 10 1/3 = -1/3(x - 1)

y - 31/3 = -x/3 + 1/3

For Line B, we have:

y - y₁ = m(x - x₁)

y - (-2) = 1/3(x - (-34))

y + 2 = x/3 + 34/3

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Answer:

------------------------

We know the property of an absolute value: it is never negative.

Hence the given inequality is equivalent to below equation:

This is the only solution.

When we solve for K in the inequality, 0 ≥ |(3K + 1) / 74|, the result obtained is -1/3

We can solve the expression 0 ≥ |(3K + 1) / 74| as illustrated below:

0 ≥ |(3K + 1) / 74|

Remove the absolute sign

0 ≥ (3K + 1) / 74

Cross multiply

0 ≥ (3K + 1) / 74

0 × 74 ≥ 3K + 1

0 ≥ 3K + 1

Collect like terms

0 - 1 ≥ 3K

-1 ≥ 3K

Divide both sides by 3

-1/3 ≥ K

K = -1/3

Thus, we can conclude from the above calculation that the value of K in the inequality, 0 ≥ |(3K + 1) / 74| is -1/3

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The exact area of our rectangle is 2 square units

The total area of the rectangle can be found by adding up the area of each square. We have four unit squares and one half-square, which can be represented as 4 + 1/2. To add these two values, we need to find a common denominator, which is 2.

Thus, we can represent the area of the rectangle as

=> 8/2 + 1/2 = 9/2 square units.

We know that the area of a rectangle is calculated by multiplying its length by its width. Thus, we can represent the area of our rectangle as follows:

A = lw

where A is the area, l is the length, and w is the width.

Now, we need to differentiate this equation with respect to the width w. This means we are finding the rate of change of the area with respect to the width. Using the product rule of differentiation, we get:

dA/dw = l * dw/dw + w * dl/dw

Since the width is constant (it does not change), the second term on the right-hand side of the equation is zero. Thus, we are left with:

dA/dw = l

To calculate the exact area of our rectangle, we can use the concept of limits. We can start by approximating the area of our rectangle with a width of 1 square unit. In this case, the area of the rectangle would be 4 square units.

We can then approach the width of 1/2 of a square unit by taking the limit as the width approaches 1/2. Using the derivative we calculated earlier, we can represent this limit as follows:

lim(w→1/2) A = lim(w→1/2) lw = l * lim(w→1/2) w = 4 * (1/2) = 2 square units

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Out of the students surveyed, 90 played a musical instrument.

To find the total number of students who played a musical instrument, we need to look at the table provided and sum up the number of boys and girls who played an instrument.

From the table, we can see the following:
- Boys who played an instrument: 42
- Girls who played an instrument: 48

To find the total number of students who played a musical instrument, simply add the number of boys and girls together:

Total students who played an instrument = (Number of boys who played an instrument) + (Number of girls who played an instrument)

Total students who played an instrument = 42 (boys) + 48 (girls)

Total students who played an instrument = 90

So, of the students surveyed, 90 played a musical instrument.

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the car is approximately 6.6 years old. The closest option provided is 6 years, so the answer is (C) 6 years.

Given:

y = A(1 – r)t

The value of a car is half what it originally cost, which means:

y = 1/2 A

The rate of depreciation is 10%, which means:

r = 0.1

Substituting these values in the equation, we get:

1/2 A = A(1 – 0.1)t

Simplifying, we get:

1/2 = 0.9t

Solving for t, we get:

t = ln(1/2) / ln(0.9) ≈ 6.6 years

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The area of the lawn that is watered by the sprinkler is approximately 200.96 square feet.

The area of the field that's  doused  by the sprinkler is a sector of a circle with a compass of 8  bases and a central angle of 360 degrees.

To find the area of the sector, we can use the formula

Area = ( θ/ 360) x πr2

 where θ is the central angle in degrees,

r is the radius of the circle,

and π is the constant pi.  

Substituting the given values,

we get   Area = (360/360) x3.14 x 82

Area = 3.14 x 64  Area = 200.96

Thus, the area of the field that's  doused  by the sprinkler is  roughly 200.96 square feet.

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The complete question is as follows:

Michael is using a rotating sprinkler to water his lawn. The sprinkler rotates in a complete circle. It sprays water at most 8 feet. Find the area of the lawn that is watered.

Answer:

y-3x-2

Step-by-step explanation:

take 2 points (1,1), (3,7)

find the slope: 7-1/3-1=3

plug into y=2x+b (used pt (1,1) )

1=3(1)+b

1=3+b

b=-2

y=3x-2

a) The quantity (in hundreds of units) that will give maximum profit: [tex]x^{2}[/tex] + 6x - 3022 = 0

b) The maximum profit is approximately $202,573.42.

An equation is a mathematical statement that shows that two expressions are equal. It typically contains variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

(a) To find the quantity that will give maximum profit, we need to find the level of output at which marginal revenue equals marginal cost.

The total revenue function for the monopoly is TR = px, where p is the price and x is the quantity sold. The marginal revenue is the derivative of the total revenue with respect to x, which is MR = d(TR)÷dx = p + x(dp÷dx).

To find the marginal revenue function, we differentiate the demand function with respect to x:

dp÷dx = -(2÷3)x

So, the marginal revenue function is:

MR = (2104 - (1÷3)[tex]x^{2}[/tex]) - (2÷3)[tex]x^{2}[/tex]

To find the marginal cost function, we differentiate the average cost function with respect to x:

dC÷dx = 22 + 2x

So, the marginal cost function is:

MC = 22 + 2x

To find the level of output at which MR = MC, we set the two functions equal to each other:

(2104 - (1÷3)[tex]x^{2}[/tex]) - (2÷3)[tex]x^{2}[/tex] = 22 + 2x

Simplifying this equation, we get:

[tex]x^{2}[/tex] + 6x - 3022 = 0

(b) Using the quadratic formula, we find that:

x = (-6 ± √( 36- 4(1)(-3022))) / 2(1)

x = (-6 ± √(36444)) / 2

x = (-6 ± 190.81) ÷ 2

x ≈ -98.4 or x ≈ 92.4

Since we can't produce a negative quantity, we choose x ≈ 92.4 as the quantity that will give maximum profit.

Therefore, the level of output that will maximize profit is 924 units (in hundreds of units).

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The opening of the pail has a diameter of approximately 55.4 inches (since it's twice the diameter of the base).

Based on the information you provided, it sounds like you have some great supplies for making sand castles at the beach with your friends.

Firstly, let's take a look at the pail. You mentioned that it has a base with a circumference of 87 inches, which means that the diameter of the base is approximately 27.7 inches (since circumference = pi x diameter). The pail is also 12 inches tall and has an opening on top that is twice the diameter of the base. Therefore, the opening has a diameter of approximately 55.4 inches (since it's twice the diameter of the base). With these measurements, you can use the pail to make some pretty big sand castles!

Next, you mentioned a plastic pyramid mold that has a square base with an edge that measures 6 inches and is 7 inches tall. This mold should be perfect for making pyramid-shaped sand castles. Just fill it with sand, pack it down, and carefully remove the mold to reveal your pyramid!

Finally, you mentioned an empty soup can with a diameter of 5.25 inches and is 6.5 inches tall. This can could be used to make cylindrical shapes in your sand castles. Simply pack sand around the can, press down firmly, and carefully remove the can to reveal your cylinder.

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Answer:

The Answer is 5

Step-by-step explanation:

Pink beads : White beads

30 : 35

( 30 / 5 ) : ( 35 / 5 )

6 : 7

To find the number of beads, You divide the number of pink beads available by the number of pink beads in the ratio:

= 30 / 6

= 5 necklaces

hope this helps :)

It should be noted that with the equations 6 old printers would produce a batch of comic books in less time.

Using 2 new printers is 27 minutes faster than using 6 old printers.

6 old printers would produce 6*(1/150) = 2/75 of a batch per minute.

2 new printers would produce 2*(1/72) = 1/36 of a batch per minute.

(5/6) * t = 30

(4/2) * t' = 18

Solving for t and t', we get:

t = 36 minutes

t' = 9 minutes

Therefore, using 6 old printers would take 36 minutes to produce a batch of comic books, while using 2 new printers would take only 9 minutes. Using 2 new printers is 27 minutes faster than using 6 old printers.

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Therefore, Tariq has collected 201.09 g trace element.

We are given that Tariq uses a container that is in the shape of a cylinder. The radius of the cylinder is 7.6 cm and the height of the cylinder is 17 cm.

Firstly, we will find the volume of the cylinder by applying the formula

The volume of the cylinder = [tex]\pi r^{2} h[/tex]

As we know the values of r and h, we will substitute these values in the given formula

The volume of cylinder = 3.14 * 17.6 * 7.6 *7.6

The volume of cylinder = 55.264 * 57.76

Volume = 3192.04 [tex]cm^{3}[/tex]

We are given that the water sample shows 0.063 grams of some trace element for every cubic centimeter of water. Therefore, we will multiply the volume of the cylinder by 0.063 to calculate the number of trace elements collected.

= 3192.04 * 0.063

= 201.09

Therefore, a total of 201.09 g of trace elements was collected by Tariq.

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270 students in a seventh-grade class of 450 would have a cell phone which denotes three-fifths of seventh graders using fractions.

Total number of students = 450

Percent of students who have cell phones = 3/5 th

In a class, if there are three-fifths of students have cell phones, that means we need to calculate the remaining percent of students who did not have cell phones.

Students without cell phones = 1 - 3/5 = 2/5

The total number of students with cell phones = (3/5) x 450

The total number of students with cell phones = 270

Therefore, we can conlcude that 270 students in a seventh-grade class of 450 would have a cell phone.

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Triangle A′B′C′ is a dilation of ​triangle ABC​ about point P with a scale factor of 1/2. The dilation is a reduction.

A dilation is a transformation that changes the size of a figure, but not its shape. It can be either an enlargement or a reduction depending on the value of the scale factor.

If the scale factor is greater than 1, then the image will be larger than the original figure. This is called an enlargement.

If the scale factor is between 0 and 1, then the image will be smaller than the original figure. This is called a reduction.

In this case, the scale factor is 1/2, which is less than 1. Therefore, the image of triangle ABC is smaller than the original triangle, and the dilation is a reduction.

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