Question

The flight of a soccer ball when kicked follows the function y = –0.01x(x - 115) where x is the horizontal distance in feet and y is the height in feet above the ground for the ball.

a) How far does the soccer ball travel?
b) What is the max height of the ball?

Answer 1

The soccer ball travels a horizontal distance of 115 feet before hitting the ground, and the maximum height it reaches is 32.8125 feet.

Step-by-step explanation:

The student's question involves the function y = –0.01x(x - 115) that represents the flight of a soccer ball when kicked. To find how far the soccer ball travels, we need to determine when the ball reaches the ground again, which occurs when y equals zero. Setting the function equal to zero and solving, we get:

y = -0.01x(x - 115) = 0

This quadratic equation has two solutions for x: x = 0 and x = 115. The ball is initially at x = 0, so the distance it travels horizontally before it hits the ground is x = 115 feet.

To determine the maximum height of the ball, we need to find the vertex of the parabola formed by the quadratic function. The horizontal distance at the vertex is given by x = -b/(2a), for the general quadratic equation ax2 + bx + c. Here, a = -0.01 and b = 1.15, so the maximum height occurs at x = 115/2 = 57.5 feet. Plugging x = 57.5 back into the function gives:

y = -0.01 * 57.5(57.5 - 115) = 32.8125 feet

So, the maximum height the ball reaches is 32.8125 feet.