Question
A soccer ball is kicked from the ground at an upward velocity of 25 feet per second. This situation can be represented by the function f(x)=−16x2+25x
. Graph the function. Identify the x-intercept(s) and interpret its meaning.(1 point)
Responses
(0,1.563)
; The soccer ball will return to the ground 1.563 seconds after it has been kicked.
left parenthesis 0 comma 1.563 right parenthesis ; The soccer ball will return to the ground 1.563 seconds after it has been kicked.
(1.563,0)
; The soccer ball will return to the ground 1.563 seconds after it has been kicked.
left parenthesis 1.563 comma 0 right parenthesis ; The soccer ball will return to the ground 1.563 seconds after it has been kicked.
(0.781,9.766)
; The soccer ball will reach a maximum height of 9.766 feet 0.781 seconds after being kicked.
left parenthesis 0.781 comma 9.766 right parenthesis ; The soccer ball will reach a maximum height of 9.766 feet 0.781 seconds after being kicked.
(1.25,0)
; The soccer ball will return to the ground 1.25 seconds after it has been kicked.
. Graph the function. Identify the x-intercept(s) and interpret its meaning.(1 point)
Responses
(0,1.563)
; The soccer ball will return to the ground 1.563 seconds after it has been kicked.
left parenthesis 0 comma 1.563 right parenthesis ; The soccer ball will return to the ground 1.563 seconds after it has been kicked.
(1.563,0)
; The soccer ball will return to the ground 1.563 seconds after it has been kicked.
left parenthesis 1.563 comma 0 right parenthesis ; The soccer ball will return to the ground 1.563 seconds after it has been kicked.
(0.781,9.766)
; The soccer ball will reach a maximum height of 9.766 feet 0.781 seconds after being kicked.
left parenthesis 0.781 comma 9.766 right parenthesis ; The soccer ball will reach a maximum height of 9.766 feet 0.781 seconds after being kicked.
(1.25,0)
; The soccer ball will return to the ground 1.25 seconds after it has been kicked.
Answers
GPT-4o mini
To analyze the function \( f(x) = -16x^2 + 25x \), which represents the height of the soccer ball as a function of time \( x \) (in seconds), we can graph the function and find its x-intercepts.
### Step 1: Finding the x-intercepts
To find the x-intercepts, we need to set \( f(x) = 0 \):
\[
-16x^2 + 25x = 0
\]
Factoring out \( x \):
\[
x(25 - 16x) = 0
\]
This gives us two solutions:
1. \( x = 0 \)
2. \( 25 - 16x = 0 \) or \( 16x = 25 \) or \( x = \frac{25}{16} = 1.5625 \)
So the x-intercepts are \( (0, 0) \) and \( (1.5625, 0) \).
### Step 2: Interpretation of x-intercepts
The x-intercepts represent the times when the soccer ball is at the ground level (height = 0).
- The intercept \( (0, 0) \) means that at time \( t = 0 \) seconds, the soccer ball is at ground level when it is kicked.
- The intercept \( (1.5625, 0) \) indicates that the soccer ball returns to ground level approximately 1.5625 seconds after being kicked.
### Conclusion
The relevant response from the options concerning the x-intercepts is:
**(1.563, 0); The soccer ball will return to the ground 1.563 seconds after it has been kicked.**
(Note: The slight difference in the time of 1.563 vs 1.5625 is due to rounding in the response options.)
### Step 1: Finding the x-intercepts
To find the x-intercepts, we need to set \( f(x) = 0 \):
\[
-16x^2 + 25x = 0
\]
Factoring out \( x \):
\[
x(25 - 16x) = 0
\]
This gives us two solutions:
1. \( x = 0 \)
2. \( 25 - 16x = 0 \) or \( 16x = 25 \) or \( x = \frac{25}{16} = 1.5625 \)
So the x-intercepts are \( (0, 0) \) and \( (1.5625, 0) \).
### Step 2: Interpretation of x-intercepts
The x-intercepts represent the times when the soccer ball is at the ground level (height = 0).
- The intercept \( (0, 0) \) means that at time \( t = 0 \) seconds, the soccer ball is at ground level when it is kicked.
- The intercept \( (1.5625, 0) \) indicates that the soccer ball returns to ground level approximately 1.5625 seconds after being kicked.
### Conclusion
The relevant response from the options concerning the x-intercepts is:
**(1.563, 0); The soccer ball will return to the ground 1.563 seconds after it has been kicked.**
(Note: The slight difference in the time of 1.563 vs 1.5625 is due to rounding in the response options.)