throwing a softball

Question

Adela throws a softball straight into the air. The ball leaves her hand when it is exactly 5 ft from the ground. The height h of a ball, in feet, can be written as a function of time t, in seconds, as h(t)=-16t² + 40t + 5. What is the maximum height the ball reaches? What time does it reach that maximum height? What time does the softball hit the ground?

Answer 1

Hi Haley,

 

Let's begin by writing the quadratic function in standard form

(ax² + bx + c):

    h(t) = -16t² + 40t + 5 (a = -16; b = 40; c =5)

 

Next, let's find the time (t) needed for the ball to reach its maximum height using the axis of symmetry formula (x = -b/2a) for a parabola:

    t = -40/[2(-16)]

    t = 40/32

    t = 5/4

    t = 1.25 seconds to reach maximum height

 

Next, let's plug in the value of 1.25 for t into the height function, h(t), to calculate the maximum height of the ball:

    h(t) = -16(1.25)² + 40(1.25) + 5

    h(t) = -16(1.5625) + 50 + 5

    h(t) = -25 + 55

    h(t) = 30 feet (maximum height)

 

Finally, let's set the height function, h(t), to zero and determine the time (t) when the ball hits the ground:

    -16t² + 40t + 5 = 0

    -16t² + 40t = -5

    (-16t²)/(-16) + 40t/(-16) = -5/(-16)

    t² - 2.5t = 0.3125

    t² - 2.5t + [(0.5)(2.5)]² = 0.3125 + [(0.5)(2.5)]²

    t² - 2.5t + (1.25)² = 0.3125 + (1.25)²

    t² - 2.5t + 1.5625 = 0.3125 + 1.5625

    (t - 1.25)² = 1.875

    t - 1.25 = +/- √1.875

    t = 1.25 +/- 1.3693

    t = 1.25 + 1.3693 (only accept positive root)

    t  = 2.6193 seconds to hit the ground

 

Below is the link to our graph of the height function, h(t):

https://dl.dropbox.com/s/5riq3ijc7jdu2j3/Graph_of_Height_Function_for_Thrown_Ball.png?raw=1

 

The graph shows three points for the height function, h(t):

    1.  y-intercept (initial height of the ball, i.e. the height from which the ball is thrown) - when t = 0, the value of h(t) is 5 (the constant term of the function). 

    2.  vertex (maximum height of the ball) - also determined algebraically above.

    3.  positive root or zero  (time at which the ball hits the ground) - also determined algebraically above.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan (Romans 5:8)