SOLUTION: 16) a ball projected vertically upward from the top of a tower has height (in feet) h(t) = -16t^2 + 64t + 80 after t seconds. At what time will the ball hit the ground?

Algebra ->  Trigonometry-basics -> SOLUTION: 16) a ball projected vertically upward from the top of a tower has height (in feet) h(t) = -16t^2 + 64t + 80 after t seconds. At what time will the ball hit the ground?       Log On


   

Question 122530: 16) a ball projected vertically upward from the top of a tower has height (in feet)
h(t) = -16t^2 + 64t + 80 after t seconds. At what time will the ball hit the ground?

a) t=2
b) t=5
c) t=4
d) t=1
e) none of these
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
When the ball hits the ground, this means h%28t%29=0 (in other words, the height is zero)

h%28t%29+=+-16t%5E2+%2B+64t+%2B+80+ Start with the given equation


0=+-16t%5E2+%2B+64t+%2B+80+ Plug in h%28t%29=0. In other words, replace h(t) with 0


Let's use the quadratic formula to solve for t:


Starting with the general quadratic

at%5E2%2Bbt%2Bc=0

the general solution using the quadratic equation is:

t+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve -16%2At%5E2%2B64%2At%2B80=0 ( notice a=-16, b=64, and c=80)




t+=+%28-64+%2B-+sqrt%28+%2864%29%5E2-4%2A-16%2A80+%29%29%2F%282%2A-16%29 Plug in a=-16, b=64, and c=80



t+=+%28-64+%2B-+sqrt%28+4096-4%2A-16%2A80+%29%29%2F%282%2A-16%29 Square 64 to get 4096



t+=+%28-64+%2B-+sqrt%28+4096%2B5120+%29%29%2F%282%2A-16%29 Multiply -4%2A80%2A-16 to get 5120



t+=+%28-64+%2B-+sqrt%28+9216+%29%29%2F%282%2A-16%29 Combine like terms in the radicand (everything under the square root)



t+=+%28-64+%2B-+96%29%2F%282%2A-16%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



t+=+%28-64+%2B-+96%29%2F-32 Multiply 2 and -16 to get -32

So now the expression breaks down into two parts

t+=+%28-64+%2B+96%29%2F-32 or t+=+%28-64+-+96%29%2F-32

Lets look at the first part:

x=%28-64+%2B+96%29%2F-32

t=32%2F-32 Add the terms in the numerator
t=-1 Divide

So one answer is
t=-1



Now lets look at the second part:

x=%28-64+-+96%29%2F-32

t=-160%2F-32 Subtract the terms in the numerator
t=5 Divide

So another answer is
t=5

So our solutions are:
t=-1 or t=5



However, since a negative time doesn't make sense, our only solution is

t=5



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Answer:


So at 5 seconds, the ball will hit the ground.