Problem solving- Complex Numbers and Roots

Question

At a carnival, A new attraction allows contestants to jump off a springboard onto a platform to be launched vertically into the air.  The object is to ring a bell located 20 feet overhead.  The distance d(t)=16t^2-bt+20, where t is the time in seconds after leaving the platform, and b is the takeoff velocity from the platform. 

 

1.  Kate watches some of the contestants. She theorizes that if the platform lanches a contestant with a takeoff velocity of at least 32 ft./s the contestant can ring the bell.

a. Find the zeros for the function using 32 feet per second as the takeoff velocity. 

B. Is Kate's theory valid?

Answer 1

I suspect the +16 is a typo, unless it is intentional and part of the question. The -16 comes from

d(t) = (1/2)at² + v₀t + d₀

a = -g = -32 ft/s²

v₀ = initial vertical velocity imparted by the platform, v₀ positive up

d₀ = height of platform (assume it is zero?)

Then

d(t) = -16t² + v₀t

Assume v₀ = 32 ft/s, and you want to reach a height of 20 ft. Is there a time t when this is realized?

20 = -16t² + 32t

-16t² + 32t - 20 = 0

Multiply both sides by -1 and get

16t² - 32t + 20 = 0

Divide by 4 to simplify a little.

4t² - 8t + 5 = 0

t = [8 ± √(64 - 4*4*5)] / 8

Note you get a complex number, so a launch speed of 32 ft/s will not get you up to 20 ft.