Nate tosses a ball up a hill for his dog to chase. The path of the ball is modeled by the function y=- 1 4 x^ 2 + 33 5 x. , where x is the ball's horizontal distance from Nate in feet and y is the ball's height in feet. The hill is modeled by the line y = 1/5 * x . How far does the ball travel horizontally before it hits the ground? (in feet, round to tenth place if necessary )

The ball's horizontal distance from Nate in feet when the ball hits the ground is 25.6 feet

Step-by-step explanation:

The given equation that represents the path of the ball is y = -1/4·x² + 33/5·x

The equation of the that models the hill is y = 1/5 × x  = 1/5·x

Where, x represents the ball's horizontal distance from Nate in feet and y represents the ball's height in feet

To find the point of contact, we equate both equations to find the common solutions as follows;

-1/4·x² + 33/5·x = 1/5·x

-1/4·x² + 33/5·x - 1/5·x = -1/4·x² + 32/5·x = 0

-1/4·x² + 32/5·x = 0

0 = 1/4·x² - 32/5·x

1/4·x² - 32/5·x = 0

x·(1/4·x - 32/5) = 0

x = 0 or 1/4·x - 32/5, and x = 32/5 × 4 = 128/5 = 25.6

The ball's horizontal distance from Nate in feet = x = 25.6 feet

The ball's horizontal distance from Nate in feet when the ball hits the ground is 25.6 feet.