Let's analyze the function \( h(x) = -130x(x-60) \), which describes the height of the football as a function of the horizontal distance traveled, \( x \).
1) What is the y-intercept?
The y-intercept occurs when \( x = 0 \). To find the y-intercept, we will substitute \( x = 0 \) into the height function:
\[ h(0) = -130(0)(0-60) = -130(0)(-60) = 0 \]
Thus, the y-intercept is \( (0, 0) \).
2) At what time is the football at the highest height?
To find the highest point of the parabola represented by \( h(x) \), we can find the vertex. The function is a downward-opening parabola (since the coefficient of \( x^2 \) is negative).
The x-coordinate of the vertex (the maximum height) can be found using the formula:
\[ x = -\frac{b}{2a} \]
In our case, we can rewrite \( h(x) \) as:
\[ h(x) = -130(x^2 - 60x) \]
Here, \( a = -130 \) and \( b = 130 \) (as we can treat \( b \) as the coefficient of \( x \)). Therefore,
\[ x = -\frac{60}{2 \times -130} = \frac{60}{260} = \frac{3}{13} \]
So, the football reaches its maximum height at \( x = \frac{3}{13} \) yards.
3) The ball will have traveled how many yards before hitting the ground?
The football will hit the ground when the height \( h(x) = 0 \). We set the height function equal to zero:
\[ -130x(x-60) = 0 \]
This gives us two factors:
- \( -130x = 0 \) → \( x = 0 \)
- \( x - 60 = 0 \) → \( x = 60 \)
So, the football will hit the ground after it has traveled \( 60 \) yards.
Summary of Answers:
- The y-intercept is \( (0, 0) \).
- The football reaches its highest height at \( x = \frac{3}{13} \) yards.
- The ball will have traveled \( 60 \) yards before hitting the ground.
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