Question

Tanisha kicks a soccer ball during a game. The height of the ball, in feet, can be modeled by the function f(x)=-16x squared +48x, where x is the time in seconds after she kicks the ball. Graph the function. Find the maximum height of the ball and how long it takes the ball to reach that height?

Answer 1

The first thing we are going to do in this case is to rewrite the function:
 f (x) = - 16x ^ 2 + 48x
 We look for the first derivative:
 f '(x) = - 32x + 48
 We match zero:
 0 = -32x + 48
 We clear x:
 32x = 48
 x = 48/32 = 24/16 = 12/8 = 6/4 = 3/2
 x = 3/2
 We verify if this value is a maximum or minimum with the second derivative:
 f '' (x) = - 32
 We evaluate in x = 3/2:
 f '' (3/2) = - 32 (<0 Is a maximum)
 Therefore the maximum height is reached in:
 x = 1.5 s
 The height is:
 f (1.5) = - 16 * (1.5) ^ 2 + 48 * (1.5)
 f (1.5) = 36 feet
 Answer:
 The maximum height of the ball is:
 36 feet
 and it takes the ball to reach that height about:
 1.5 s