Question

Can anyone help explaining step by step?

The height of a soccer ball in the air can be modeled by the equation h = -16t^2 +48f, where h is in feet of a ball that Cody kicks down the field.
How long is the ball in the air? (find the zeros).
What is the ball's max height? (find the vertex)
When will the ball reach its maximum height?

Answer 1

• 3 seconds
• (0, 0), (3, 0)
• 36 ft, (1.5, 36)

Explanation:

As the problem statement suggests, the question of time in the air can be answered by finding the zeros of the function. Similarly, the question of the ball's maximum height can be answered by finding the vertex. These findings are related by the fact that the vertex is on the line of symmetry, which passes through the midpoint between the zeros.

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(a)

The function can be factored as ...

h = -16t(t -3)

At any zero, one of the factors must be zero. (This is the "zero product rule.") The factors are zero for t=0 and for t=3.

The zeros are (0, 0) and (3, 0).

This tells you the height of the ball is zero initially (at t=0), and 3 seconds later (at t=3).

The ball is in the air for 3 seconds.

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(b)

The midpoint between the zeros is their average: (0 +3)/2 = 1.5. At that value of t, the ball will be at its maximum height.

The ball will reach its maximum height at t=1.5 seconds.

Using the above factored form to find the height, we get ...

h = -16(1.5)(1.5 -3) = 16(2.25) = 36 . . . feet

The ball's maximum height is 36 feet.

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Additional comment

The x-coordinate of the vertex of quadratic ax²+bx+c is found at the value ...

x = -b/(2a)

Here, that would be ...

t = -48/(2(-16)) = 3/2

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A graphing calculator can answer all of these questions quite rapidly.