Question

MODELING REAL LIFE A football player punts a football from a height of 2 feet. The ball reaches a maximum height of 50 feet after traveling 56 feet horizontally, and is caught 111 feet from where the ball was kicked. The path of the ball can be modeled by a parabola, where y is the height (in feet) and x is the horizontal distance traveled (in feet). At what height is the ball caught? Round your answer to the nearest tenth.

Answer 1

3.7 feet

Explanation:

You want the height at a distance of 111 feet of a ball kicked from a height of 2 feet that reaches a maximum of 50 feet at a point 56 feet horizontally from where it was kicked.

Parabola

The vertex-form equation of a parabola is ...

y = a(x -h)² +k

where the vertex is (h, k) and the scale factor is 'a'.

For the vertex (56, 50), the equation is ...

y = a(x -56)² +50

At a horizontal distance of 0 feet, we want the equation to tell us the height is 2 feet. This means ...

2 = a(0 -56)² +50

-48 = 3136a

a = -48/3136 = -3/196

Caught

The height at a distance of 111 feet is ...

y = -3/196(111 -56)² +50 = -3/196(3025) +50 = 3 137/196 ≈ 3.7 . . . feet

The ball is caught at a height of 3.7 feet.

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