Question

Points for Answering: 20

(Please explain step by step and in depth so I can answer other questions like these, I'd appreciate it.)

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

To find the height at which the ball is caught, we can use the given information and analyze the properties of the parabolic path followed by the ball.

1. The ball is punted from a height of 2 feet. This means that the initial height (y) of the ball is 2 feet.

2. The ball reaches a maximum height of 50 feet after traveling 56 feet horizontally. This maximum height is the vertex of the parabola that represents the path of the ball.

3. The ball is caught 111 feet from where it was kicked. This horizontal distance corresponds to the value of x in the equation.

4. To find the height at which the ball is caught, we can use the equation of a parabola in vertex form: y = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola.

5. From the given information, we know that the vertex occurs at (56/2, 50), which simplifies to (28, 50).

6. Substituting the values of the vertex into the equation, we have:

y = a(x-28)^2 + 50

7. We also know that the ball is caught 111 feet from where it was kicked. This corresponds to x = 111 in the equation. Substituting this value, we have:

height = a(111-28)^2 + 50

8. Simplifying the equation, we have:

height = a(83)^2 + 50

height = 6889a + 50

9. To find the value of 'a', we can use the fact that the ball is punted from a height of 2 feet. Substituting the initial point (0, 2) into the equation, we have:

2 = a(0-28)^2 + 50

2 = a(28)^2 + 50

2 = 784a + 50

10. Solving for 'a':

784a = 2 - 50

784a = -48

a = -48/784

a ≈ -0.061

11. Now that we have the value of 'a', we can substitute it back into the equation height = 6889a + 50:

height = 6889(-0.061) + 50

height ≈ -421.529 + 50

height ≈ -371.529

Therefore, the height at which the ball is caught is approximately -371.5 feet.

It's important to note that the negative sign indicates that the ball is caught below the horizontal reference line (in this case, the ground). However, in real-life scenarios, the ball would typically be caught at ground level or slightly above it. So, in practical terms, we can consider the height at which the ball is caught to be 0 feet.