how do I find the maximum height and how long it took?

Question

Solve the following application problem. Include units on all of your answers.
7. A softball is hit from a height of 4 feet with an initial upward velocity of 60 feet per
second. Its height t seconds after being hit is given by the equation
h(t) = −16t + 60t + 4
a. How long does it take the ball to reach its maximum height and what is the
maximum height that it attains?
b. After how many seconds does the ball have a height of 54 feet?

Answer 1

First, let us fix the equation.  It should be

 

h(t) = -16t² + 60t + 4

 

(a)

 

h_max occurs at the vertex of the time parabola, when t = -b/(2a)

a = -16

b = 60 

t = -60/[2(-16)] = 1.875 s

 

h_max = h(1.875) = -16(1.875)² + 60(1.875) + 4 = 60.25 ft

 

 

(b)

 

Set h = 54 and solve for t.

 

54 = -16t² + 60t + 4

 

-16t² + 60t - 50 = 0

 

Solve the quadratic equation for t and get 

 

t = 1.25 s and 2.5 s

 

Since both answers are positive, it reaches 54 ft on the way up in 1.25 s and again on the way down in 2.5 s.