1 answer
(click or scroll down)
Bot GPT 4-o mini To analyze the path of the football described by the function \( h(x) = -130x(x - 60) \), we will work through the questions one by one.
1) **What is the y-intercept? Write your answer as an ordered pair.**
The y-intercept occurs when \( x = 0 \).
\[
h(0) = -130(0)(0 - 60) = 0
\]
Therefore, the y-intercept is \( (0, 0) \).
2) **At what time is the football at the highest height?**
The function \( h(x) \) is a quadratic equation which opens downwards ( since the coefficient of \( x^2 \) is negative), meaning it has a maximum point (the vertex). The x-coordinate of the vertex of a quadratic function of the form \( ax^2 + bx + c \) is given by the formula:
\[
x = -\frac{b}{2a}
\]
For the function \( h(x) = -130x^2 + 7800x \):
- \( a = -130 \)
- \( b = 7800 \)
Thus,
\[
x = -\frac{7800}{2(-130)} = \frac{7800}{260} = 30
\]
Therefore, the football reaches its highest height at \( x = 30 \) yards.
3) **The ball will have traveled \_\_\_\_ yards before hitting the ground.**
The football will hit the ground when \( h(x) = 0 \). We can solve for \( x \) when \( h(x) \) equals zero:
\[
h(x) = -130x(x - 60) = 0
\]
Setting each factor to zero gives:
\[
-130x = 0 \quad \Rightarrow \quad x = 0
\]
\[
x - 60 = 0 \quad \Rightarrow \quad x = 60
\]
Therefore, the ball will hit the ground after traveling \( 60 \) yards.
To summarize the answers:
1) The y-intercept is: \( (0, 0) \)
2) The football reaches its highest height at: \( 30 \) yards
3) The ball will have traveled: \( 60 \) yards before hitting the ground.