Question
Question 1(Multiple Choice Worth 1 points)
(08.04 LC)
Sloane kicked a soccer ball off the ground at a speed of 36 feet per second. The height of the ball can be represented by the function H(t) = −16t2 + 36t. How many seconds did the ball travel before returning to the ground?
t = 0.44 seconds
t = 2.25 seconds
t = 16 seconds
t = 36 seconds
Question 2(Multiple Choice Worth 1 points)
(08.04 MC)
The graph of the function C(x) = −0.34x2 + 12x + 62 is shown. The function models the production cost, C, in thousands of dollars for a tire company to manufacture a tire, where x is the number of tires produced, in thousands:
graph of a parabola opening down passing through points negative 4 and 57 hundredths comma zero, zero comma 62, 1 and 12 hundredths comma 75, 17 and 65 hundredths comma 167 and 55 hundredths, 34 and 18 hundredths comma 75, and 39 and 87 hundredths comma zero
If the company wants to keep its production costs under $75,000, then which constraint is reasonable for the model?
−4.57 ≤ x ≤ 39.87
1.12 ≤ x ≤ 34.18
−4.57 ≤ x ≤ 1.12 and 34.18 ≤ x ≤ 39.87
0 ≤ x < 1.12 and 34.18 < x ≤ 39.87
Question 3(Multiple Choice Worth 1 points)
(08.04 MC)
A particle is moving along a projectile path at an initial height of 160 feet with an initial speed of 144 feet per second. This can be represented by the function H(t) = −16t2 + 144t + 160. What is the maximum height of the particle?
128 feet
224 feet
272 feet
484 feet
Question 4(Multiple Choice Worth 1 points)
(08.04 MC)
Marissa is painting her rectangular patio, with the exception of a bench that does not need to be painted:
rectangle with a length of x plus 15 and width of x plus 10 with a rectangle in the bottom right corner labeled bench that has a length of 5 and width of 1
Write an equation to determine the area, A, of the patio that will be painted.
A = (x + 20)(x + 11)
A = (x + 15)(x + 10) + 5
A = (x + 15)(x + 10) − 5
A = (x + 9)(x + 10)
Question 5(Multiple Choice Worth 1 points)
(08.04 MC)
Samar serves a volleyball over a net at an initial height of 5 feet with a velocity of 100 feet per second. This can be represented by the function H(t) = −16t2 + 100t + 5. If the other team misses, about how long does it take the ball to hit the ground?
0.05 seconds
1.59 seconds
6.30 seconds
7.89 seconds
Question 6(Multiple Choice Worth 1 points)
(08.04 MC)
The graph of the function P(x) = −0.52x2 + 23x + 92 is shown. The function models the profits, P, in thousands of dollars for a candle company to manufacture a candle, where x is the number of candles produced, in thousands:
graph of a parabola opening down passing through points negative 3 and 69 hundredths comma zero, zero comma 92, 6 and 84 hundredths comma 225, 22 and 11 hundredths comma 346 and 33 hundredths, 37 and 39 hundredths comma 225, and 47 and 92 hundredths comma zero
If the company wants to keep its profits at or above $225,000, then which constraint is reasonable for the model?
0 ≤ x < 6.84 and 37.39 < x ≤ 47.92
−3.69 ≤ x ≤ 6.84 and 37.39 < x ≤ 47.92
6.84 ≤ x ≤ 37.39
−3.69 ≤ x ≤ 47.92
Question 7(Multiple Choice Worth 1 points)
(08.04 MC)
Ken wants to build a table and put a border around it. The table and border must have an area of 3,276 square inches. The table is 36 inches wide and 72 inches long without the border. Which quadratic equation can be used to determine the thickness of the border, x?
4x2 + 216x + 2,592 = 0
4x2 + 216x − 684 = 0
2x2 + 216x − 3,276 = 0
x2 + 108x + 3,276 = 0
Question 8(Multiple Choice Worth 1 points)
(08.04 MC)
A local school needs to paint the floor of its theater room, where the length of the floor, x, is at least 12 feet. The width of the floor is 4 feet less than the length. It will have a stage and a closet, and the remaining area of the floor will be painted. The dimensions are shown in the diagram:
rectangle with length of x ft and width of x minus 4 ft, right triangle inside labeled stage with height of x minus 4 ft and base of 8 ft, rectangle inside labeled closet with length of 8 ft and width of 4 ft, the rest of the rectangle is labeled floor
Let A represent the painted area, in square feet, of the floor. Choose the correct equation to solve for area (A).
A = x(x − 4) − 0.5(8)(x − 4) − 4(8)
A = x(x − 4) − (8)(x − 4) − 4(8)
A = x(x − 4) + (8)(x − 4) + 4(8)
A = x(x − 4) + 0.5(8)(x − 4) + 4(8)
Question 9(Multiple Choice Worth 1 points)
(08.04 MC)
For the graph, what is a reasonable constraint so that the function is at least 200?
graph of y equals minus 2 times the square of x plus 30 times x plus 200
−5 ≤ x ≤ 20
0 ≤ x ≤ 15
x ≥ 0
All real numbers
Question 10(Multiple Choice Worth 1 points)
(08.04 MC)
The length of a rectangular frame is represented by the expression 2x + 10, and the width of the rectangular frame is represented by the expression 2x + 6. Write an equation to solve for the width of a rectangular frame that has a total area of 140 square inches.
4x2 + 32x − 80 = 0
4x2 + 32x + 60 = 0
2x2 + 32x − 80 = 0
x2 + 16x + 60 = 0
(08.04 LC)
Sloane kicked a soccer ball off the ground at a speed of 36 feet per second. The height of the ball can be represented by the function H(t) = −16t2 + 36t. How many seconds did the ball travel before returning to the ground?
t = 0.44 seconds
t = 2.25 seconds
t = 16 seconds
t = 36 seconds
Question 2(Multiple Choice Worth 1 points)
(08.04 MC)
The graph of the function C(x) = −0.34x2 + 12x + 62 is shown. The function models the production cost, C, in thousands of dollars for a tire company to manufacture a tire, where x is the number of tires produced, in thousands:
graph of a parabola opening down passing through points negative 4 and 57 hundredths comma zero, zero comma 62, 1 and 12 hundredths comma 75, 17 and 65 hundredths comma 167 and 55 hundredths, 34 and 18 hundredths comma 75, and 39 and 87 hundredths comma zero
If the company wants to keep its production costs under $75,000, then which constraint is reasonable for the model?
−4.57 ≤ x ≤ 39.87
1.12 ≤ x ≤ 34.18
−4.57 ≤ x ≤ 1.12 and 34.18 ≤ x ≤ 39.87
0 ≤ x < 1.12 and 34.18 < x ≤ 39.87
Question 3(Multiple Choice Worth 1 points)
(08.04 MC)
A particle is moving along a projectile path at an initial height of 160 feet with an initial speed of 144 feet per second. This can be represented by the function H(t) = −16t2 + 144t + 160. What is the maximum height of the particle?
128 feet
224 feet
272 feet
484 feet
Question 4(Multiple Choice Worth 1 points)
(08.04 MC)
Marissa is painting her rectangular patio, with the exception of a bench that does not need to be painted:
rectangle with a length of x plus 15 and width of x plus 10 with a rectangle in the bottom right corner labeled bench that has a length of 5 and width of 1
Write an equation to determine the area, A, of the patio that will be painted.
A = (x + 20)(x + 11)
A = (x + 15)(x + 10) + 5
A = (x + 15)(x + 10) − 5
A = (x + 9)(x + 10)
Question 5(Multiple Choice Worth 1 points)
(08.04 MC)
Samar serves a volleyball over a net at an initial height of 5 feet with a velocity of 100 feet per second. This can be represented by the function H(t) = −16t2 + 100t + 5. If the other team misses, about how long does it take the ball to hit the ground?
0.05 seconds
1.59 seconds
6.30 seconds
7.89 seconds
Question 6(Multiple Choice Worth 1 points)
(08.04 MC)
The graph of the function P(x) = −0.52x2 + 23x + 92 is shown. The function models the profits, P, in thousands of dollars for a candle company to manufacture a candle, where x is the number of candles produced, in thousands:
graph of a parabola opening down passing through points negative 3 and 69 hundredths comma zero, zero comma 92, 6 and 84 hundredths comma 225, 22 and 11 hundredths comma 346 and 33 hundredths, 37 and 39 hundredths comma 225, and 47 and 92 hundredths comma zero
If the company wants to keep its profits at or above $225,000, then which constraint is reasonable for the model?
0 ≤ x < 6.84 and 37.39 < x ≤ 47.92
−3.69 ≤ x ≤ 6.84 and 37.39 < x ≤ 47.92
6.84 ≤ x ≤ 37.39
−3.69 ≤ x ≤ 47.92
Question 7(Multiple Choice Worth 1 points)
(08.04 MC)
Ken wants to build a table and put a border around it. The table and border must have an area of 3,276 square inches. The table is 36 inches wide and 72 inches long without the border. Which quadratic equation can be used to determine the thickness of the border, x?
4x2 + 216x + 2,592 = 0
4x2 + 216x − 684 = 0
2x2 + 216x − 3,276 = 0
x2 + 108x + 3,276 = 0
Question 8(Multiple Choice Worth 1 points)
(08.04 MC)
A local school needs to paint the floor of its theater room, where the length of the floor, x, is at least 12 feet. The width of the floor is 4 feet less than the length. It will have a stage and a closet, and the remaining area of the floor will be painted. The dimensions are shown in the diagram:
rectangle with length of x ft and width of x minus 4 ft, right triangle inside labeled stage with height of x minus 4 ft and base of 8 ft, rectangle inside labeled closet with length of 8 ft and width of 4 ft, the rest of the rectangle is labeled floor
Let A represent the painted area, in square feet, of the floor. Choose the correct equation to solve for area (A).
A = x(x − 4) − 0.5(8)(x − 4) − 4(8)
A = x(x − 4) − (8)(x − 4) − 4(8)
A = x(x − 4) + (8)(x − 4) + 4(8)
A = x(x − 4) + 0.5(8)(x − 4) + 4(8)
Question 9(Multiple Choice Worth 1 points)
(08.04 MC)
For the graph, what is a reasonable constraint so that the function is at least 200?
graph of y equals minus 2 times the square of x plus 30 times x plus 200
−5 ≤ x ≤ 20
0 ≤ x ≤ 15
x ≥ 0
All real numbers
Question 10(Multiple Choice Worth 1 points)
(08.04 MC)
The length of a rectangular frame is represented by the expression 2x + 10, and the width of the rectangular frame is represented by the expression 2x + 6. Write an equation to solve for the width of a rectangular frame that has a total area of 140 square inches.
4x2 + 32x − 80 = 0
4x2 + 32x + 60 = 0
2x2 + 32x − 80 = 0
x2 + 16x + 60 = 0
Answers
Answered by
GPT-4o mini
Here are the answers to your questions based on the information provided:
**Question 1**: To find how long the ball traveled before returning to the ground, you need to find when \( H(t) = 0 \). The equation is \( -16t^2 + 36t = 0 \). Factoring gives us:
\[
t(36 - 16t) = 0
\]
Which gives \( t = 0 \) or \( t = \frac{36}{16} = 2.25 \) seconds.
**Answer**: **t = 2.25 seconds**
---
**Question 2**: You want to keep the production costs under $75,000, which translates to \( C(x) < 75 \). Find the x-values where \( C(x) = 75 \):
\[
-0.34x^2 + 12x + 62 = 75
\]
This simplifies to:
\[
-0.34x^2 + 12x - 13 = 0
\]
Using the graph, the reasonable x-values for costs under $75,000 would be between \( 1.12 \) and \( 34.18 \).
**Answer**: **1.12 ≤ x ≤ 34.18**
---
**Question 3**: To find the maximum height, use the vertex formula \( t = -\frac{b}{2a} \) for the function \( H(t) = -16t^2 + 144t + 160 \):
\[
t = -\frac{144}{2(-16)} = \frac{144}{32} = 4.5
\]
Plug this back into \( H(t) \) to find the maximum height:
\[
H(4.5) = -16(4.5)^2 + 144(4.5) + 160 = 324 + 160 = 484
\]
**Answer**: **484 feet**
---
**Question 4**: The area to be painted is the area of the whole rectangle minus the area of the bench. Thus:
\[
A = (x + 15)(x + 10) - (5 \cdot 1)
\]
This is:
\[
A = (x + 15)(x + 10) - 5
\]
**Answer**: **A = (x + 15)(x + 10) − 5**
---
**Question 5**: Set \( H(t) = 0 \):
\[
-16t^2 + 100t + 5 = 0
\]
Using the quadratic formula:
\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-100 \pm \sqrt{10000 + 320}}{-32}
\]
This gives approximately \( 6.30 \) seconds.
**Answer**: **6.30 seconds**
---
**Question 6**: For profits at or above $225,000, set \( P(x) ≥ 225 \):
\[
-0.52x^2 + 23x + 92 = 225
\]
This leads to:
\[
-0.52x^2 + 23x - 133 = 0
\]
Using the graph, it is reasonable that profits stay between:
\[ 0 ≤ x < 6.84 \text{ and } 37.39 < x ≤ 47.92 \]
**Answer**: **0 ≤ x < 6.84 and 37.39 < x ≤ 47.92**
---
**Question 7**: To find the total area with the border:
\[
\text{Total Area} = (36 + 2x)(72 + 2x)
\]
Set this equal to \( 3276 \):
\[
(36 + 2x)(72 + 2x) = 3276
\]
Expanding gives a quadratic equation:
\[
4x^2 + 216x - 684 = 0
\]
**Answer**: **4x² + 216x − 684 = 0**
---
**Question 8**: The area that will be painted:
\[
A = x(x - 4) - \left(\frac{1}{2}(8)(x - 4) + 4(8)\right)
\]
This is:
\[
A = x(x - 4) - 0.5(8)(x - 4) - 4(8)
\]
**Answer**: **A = x(x − 4) − 0.5(8)(x − 4) − 4(8)**
---
**Question 9**: To maintain a function at least 200, it would typically depend on where the graph intersects or remains above 200, thus:
**Answer**: **x ≥ 0**
---
**Question 10**: To find the area of the frame:
\[
\text{Area} = \text{Length} \times \text{Width} = (2x + 10)(2x + 6) = 140
\]
This leads to:
\[
4x^2 + 32x - 80 = 0
\]
**Answer**: **4x² + 32x − 80 = 0**
---
Feel free to ask for more details on any specific question.
**Question 1**: To find how long the ball traveled before returning to the ground, you need to find when \( H(t) = 0 \). The equation is \( -16t^2 + 36t = 0 \). Factoring gives us:
\[
t(36 - 16t) = 0
\]
Which gives \( t = 0 \) or \( t = \frac{36}{16} = 2.25 \) seconds.
**Answer**: **t = 2.25 seconds**
---
**Question 2**: You want to keep the production costs under $75,000, which translates to \( C(x) < 75 \). Find the x-values where \( C(x) = 75 \):
\[
-0.34x^2 + 12x + 62 = 75
\]
This simplifies to:
\[
-0.34x^2 + 12x - 13 = 0
\]
Using the graph, the reasonable x-values for costs under $75,000 would be between \( 1.12 \) and \( 34.18 \).
**Answer**: **1.12 ≤ x ≤ 34.18**
---
**Question 3**: To find the maximum height, use the vertex formula \( t = -\frac{b}{2a} \) for the function \( H(t) = -16t^2 + 144t + 160 \):
\[
t = -\frac{144}{2(-16)} = \frac{144}{32} = 4.5
\]
Plug this back into \( H(t) \) to find the maximum height:
\[
H(4.5) = -16(4.5)^2 + 144(4.5) + 160 = 324 + 160 = 484
\]
**Answer**: **484 feet**
---
**Question 4**: The area to be painted is the area of the whole rectangle minus the area of the bench. Thus:
\[
A = (x + 15)(x + 10) - (5 \cdot 1)
\]
This is:
\[
A = (x + 15)(x + 10) - 5
\]
**Answer**: **A = (x + 15)(x + 10) − 5**
---
**Question 5**: Set \( H(t) = 0 \):
\[
-16t^2 + 100t + 5 = 0
\]
Using the quadratic formula:
\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-100 \pm \sqrt{10000 + 320}}{-32}
\]
This gives approximately \( 6.30 \) seconds.
**Answer**: **6.30 seconds**
---
**Question 6**: For profits at or above $225,000, set \( P(x) ≥ 225 \):
\[
-0.52x^2 + 23x + 92 = 225
\]
This leads to:
\[
-0.52x^2 + 23x - 133 = 0
\]
Using the graph, it is reasonable that profits stay between:
\[ 0 ≤ x < 6.84 \text{ and } 37.39 < x ≤ 47.92 \]
**Answer**: **0 ≤ x < 6.84 and 37.39 < x ≤ 47.92**
---
**Question 7**: To find the total area with the border:
\[
\text{Total Area} = (36 + 2x)(72 + 2x)
\]
Set this equal to \( 3276 \):
\[
(36 + 2x)(72 + 2x) = 3276
\]
Expanding gives a quadratic equation:
\[
4x^2 + 216x - 684 = 0
\]
**Answer**: **4x² + 216x − 684 = 0**
---
**Question 8**: The area that will be painted:
\[
A = x(x - 4) - \left(\frac{1}{2}(8)(x - 4) + 4(8)\right)
\]
This is:
\[
A = x(x - 4) - 0.5(8)(x - 4) - 4(8)
\]
**Answer**: **A = x(x − 4) − 0.5(8)(x − 4) − 4(8)**
---
**Question 9**: To maintain a function at least 200, it would typically depend on where the graph intersects or remains above 200, thus:
**Answer**: **x ≥ 0**
---
**Question 10**: To find the area of the frame:
\[
\text{Area} = \text{Length} \times \text{Width} = (2x + 10)(2x + 6) = 140
\]
This leads to:
\[
4x^2 + 32x - 80 = 0
\]
**Answer**: **4x² + 32x − 80 = 0**
---
Feel free to ask for more details on any specific question.
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