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Hassan wants to add a border to a photo he took before he frames it. The final area of the entire framed picture will be 96 squ...Question
Hassan wants to add a border to a photo he took before he frames it. The final area of the entire framed picture will be 96 square inches. The length of the picture is 10 inches, and the width is 6 inches. Which of the following is the width of the border?(1 point)
Responses
1 inch
1 inch
3 inches
3 inches
9 inches
9 inches
4 inches
May’s class is testing their egg protection contraptions by dropping them off the roof of their school. Use GeoGebra to graph the situation using the formula y=−16t2+40
. Which of the following correctly interprets the height of the school’s roof?(1 point)
Responses
41 feet
41 feet
20 feet
20 feet
40 feet
40 feet
15 feet
15 feet
Luca is in a culvert below street level. He launches an object at an upward velocity of 40 feet per second. Use GeoGebra to graph the situation using the formula y=−16t2+40t−5
. How far above or below street level is Luca’s initial launching point?(1 point)
Responses
5 feet above
5 feet above
2.5 feet below
2.5 feet below
0 feet above or below
0 feet above or below
5 feet below
Which of the following scenarios would use a quadratic equation to solve?(1 point)
Responses
The trajectory of a balloon that rises 2 feet per second is modeled to find how high the balloon will be in 5 minutes.
The trajectory of a balloon that rises 2 feet per second is modeled to find how high the balloon will be in 5 minutes.
A coach models the trajectory of a ball to find out how high the player is kicking the ball.
A coach models the trajectory of a ball to find out how high the player is kicking the ball.
An accountant models the growth of a savings account that doubles every month to find how much will be in the account in 10 years.
An accountant models the growth of a savings account that doubles every month to find how much will be in the account in 10 years.
A child places a quarter into a piggy bank every week
Use the image to answer the question.
A parabola is graphed on a coordinate plane. The x-axis ranges from negative 2 to 10 in increments of 1. The y-axis ranges from negative 8 to 3 in increments of 1.
A sports analyst graphed the trajectory of a soccer ball that was kicked by a goalkeeper. In the graph, x is the ball’s distance from the goalpost, and y is the height of the ball in meters. What was the goalkeeper’s distance from the goalpost when the ball was kicked?
(1 point)
Responses
2 meters
2 meters
6 meters
6 meters
0 meters
0 meters
4 meters
4 meters
Skip to navigation
Responses
1 inch
1 inch
3 inches
3 inches
9 inches
9 inches
4 inches
May’s class is testing their egg protection contraptions by dropping them off the roof of their school. Use GeoGebra to graph the situation using the formula y=−16t2+40
. Which of the following correctly interprets the height of the school’s roof?(1 point)
Responses
41 feet
41 feet
20 feet
20 feet
40 feet
40 feet
15 feet
15 feet
Luca is in a culvert below street level. He launches an object at an upward velocity of 40 feet per second. Use GeoGebra to graph the situation using the formula y=−16t2+40t−5
. How far above or below street level is Luca’s initial launching point?(1 point)
Responses
5 feet above
5 feet above
2.5 feet below
2.5 feet below
0 feet above or below
0 feet above or below
5 feet below
Which of the following scenarios would use a quadratic equation to solve?(1 point)
Responses
The trajectory of a balloon that rises 2 feet per second is modeled to find how high the balloon will be in 5 minutes.
The trajectory of a balloon that rises 2 feet per second is modeled to find how high the balloon will be in 5 minutes.
A coach models the trajectory of a ball to find out how high the player is kicking the ball.
A coach models the trajectory of a ball to find out how high the player is kicking the ball.
An accountant models the growth of a savings account that doubles every month to find how much will be in the account in 10 years.
An accountant models the growth of a savings account that doubles every month to find how much will be in the account in 10 years.
A child places a quarter into a piggy bank every week
Use the image to answer the question.
A parabola is graphed on a coordinate plane. The x-axis ranges from negative 2 to 10 in increments of 1. The y-axis ranges from negative 8 to 3 in increments of 1.
A sports analyst graphed the trajectory of a soccer ball that was kicked by a goalkeeper. In the graph, x is the ball’s distance from the goalpost, and y is the height of the ball in meters. What was the goalkeeper’s distance from the goalpost when the ball was kicked?
(1 point)
Responses
2 meters
2 meters
6 meters
6 meters
0 meters
0 meters
4 meters
4 meters
Skip to navigation
Answers
GPT-4o mini
Let's address each question one by one:
1. **Border Width Calculation**:
- The area of the framed picture is 96 square inches. The dimensions of the picture are 10 inches and 6 inches.
- Area of the picture: \( A = \text{length} \times \text{width} = 10 \times 6 = 60 \) square inches.
- Let \( x \) be the width of the border. Then, the dimensions of the framed picture will be \( (10 + 2x) \) and \( (6 + 2x) \).
- The area of the framed picture can be expressed as:
\[
(10 + 2x)(6 + 2x) = 96
\]
- Expanding the equation:
\[
60 + 20x + 12x + 4x^2 = 96
\]
\[
4x^2 + 32x + 60 - 96 = 0
\]
\[
4x^2 + 32x - 36 = 0
\]
\[
x^2 + 8x - 9 = 0
\]
- Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[
x = \frac{-8 \pm \sqrt{64 + 36}}{2}
\]
\[
x = \frac{-8 \pm 10}{2}
\]
- This gives us:
1. \( x = 1 \) (valid, positive border width).
2. \( x = -9 \) (not valid, negative width).
- **Answer**: **1 inch**.
2. **Height of the School’s Roof**:
- The equation \( y = -16t^2 + 40 \) describes a projectile motion where the coefficient of \( t^2 \) indicates the effect of gravity (16 feet per second squared).
- The initial height (when \( t = 0 \)) is:
\[
y = -16(0)^2 + 40 = 40
\]
- **Answer**: **40 feet**.
3. **Luca’s Initial Launching Point**:
- The formula \( y = -16t^2 + 40t - 5 \) gives Luca's position relative to street level. The value at \( t = 0 \) indicates the initial launching point:
\[
y = -16(0)^2 + 40(0) - 5 = -5
\]
- This means Luca launches 5 feet below street level.
- **Answer**: **5 feet below**.
4. **Use of Quadratic Equation**:
- A scenario involving projectile motion, like finding the height of a soccer ball, can be modeled by a quadratic equation because the motion involves a parabolic trajectory.
- The correct scenario is:
- **A coach models the trajectory of a ball to find out how high the player is kicking the ball.**
- **Answer**: **A coach models the trajectory of a ball to find out how high the player is kicking the ball.**
5. **Goalkeeper’s Distance from the Goalpost**:
- The graph would provide information where the parabola intersects the x-axis (when height is 0). This corresponds to the distance from the goalpost when the ball was kicked.
- If we assume that the y-axis intercepts the height before reaching this intersection at the point, then one possible distance can be deduced based on provided options. Typically, the x-coordinate at \( y = 0 \) is what we check.
- **Without the actual graph**, we typically choose the lowest distance based on the trajectory model—most likely:
- **Answer**: (Choose the specific value from available graph details, but likely a lower distance) Select based on graph observations.
Please let me know if you need additional help or clarifications on any of the solutions!
1. **Border Width Calculation**:
- The area of the framed picture is 96 square inches. The dimensions of the picture are 10 inches and 6 inches.
- Area of the picture: \( A = \text{length} \times \text{width} = 10 \times 6 = 60 \) square inches.
- Let \( x \) be the width of the border. Then, the dimensions of the framed picture will be \( (10 + 2x) \) and \( (6 + 2x) \).
- The area of the framed picture can be expressed as:
\[
(10 + 2x)(6 + 2x) = 96
\]
- Expanding the equation:
\[
60 + 20x + 12x + 4x^2 = 96
\]
\[
4x^2 + 32x + 60 - 96 = 0
\]
\[
4x^2 + 32x - 36 = 0
\]
\[
x^2 + 8x - 9 = 0
\]
- Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[
x = \frac{-8 \pm \sqrt{64 + 36}}{2}
\]
\[
x = \frac{-8 \pm 10}{2}
\]
- This gives us:
1. \( x = 1 \) (valid, positive border width).
2. \( x = -9 \) (not valid, negative width).
- **Answer**: **1 inch**.
2. **Height of the School’s Roof**:
- The equation \( y = -16t^2 + 40 \) describes a projectile motion where the coefficient of \( t^2 \) indicates the effect of gravity (16 feet per second squared).
- The initial height (when \( t = 0 \)) is:
\[
y = -16(0)^2 + 40 = 40
\]
- **Answer**: **40 feet**.
3. **Luca’s Initial Launching Point**:
- The formula \( y = -16t^2 + 40t - 5 \) gives Luca's position relative to street level. The value at \( t = 0 \) indicates the initial launching point:
\[
y = -16(0)^2 + 40(0) - 5 = -5
\]
- This means Luca launches 5 feet below street level.
- **Answer**: **5 feet below**.
4. **Use of Quadratic Equation**:
- A scenario involving projectile motion, like finding the height of a soccer ball, can be modeled by a quadratic equation because the motion involves a parabolic trajectory.
- The correct scenario is:
- **A coach models the trajectory of a ball to find out how high the player is kicking the ball.**
- **Answer**: **A coach models the trajectory of a ball to find out how high the player is kicking the ball.**
5. **Goalkeeper’s Distance from the Goalpost**:
- The graph would provide information where the parabola intersects the x-axis (when height is 0). This corresponds to the distance from the goalpost when the ball was kicked.
- If we assume that the y-axis intercepts the height before reaching this intersection at the point, then one possible distance can be deduced based on provided options. Typically, the x-coordinate at \( y = 0 \) is what we check.
- **Without the actual graph**, we typically choose the lowest distance based on the trajectory model—most likely:
- **Answer**: (Choose the specific value from available graph details, but likely a lower distance) Select based on graph observations.
Please let me know if you need additional help or clarifications on any of the solutions!