SOLUTION: Science and medicine: The equation h = -16t^2 + 112t gives the height of an arrow, shot upward from the ground with an initial velocity of 112 ft/s, where t is the time after the

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Question 121298: Science and medicine:
The equation h = -16t^2 + 112t gives the height of an arrow, shot upward from the ground with an initial velocity of 112 ft/s, where t is the time after the arrow loeaves the ground. Find the time it takes for the arrow to reach a height of 180 ft.
I would really appreciate some help on this, Thank you very much.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
h+=+-16t%5E2+%2B+112t Start with the given equation


180+=+-16t%5E2+%2B+112t Plug in h=180



16t%5E2-112t%2B180=0 Get everything to the left side

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%5E2-112%2At%2B180=0 ( notice a=16, b=-112, and c=180)




t+=+%28--112+%2B-+sqrt%28+%28-112%29%5E2-4%2A16%2A180+%29%29%2F%282%2A16%29 Plug in a=16, b=-112, and c=180



t+=+%28112+%2B-+sqrt%28+%28-112%29%5E2-4%2A16%2A180+%29%29%2F%282%2A16%29 Negate -112 to get 112



t+=+%28112+%2B-+sqrt%28+12544-4%2A16%2A180+%29%29%2F%282%2A16%29 Square -112 to get 12544 (note: remember when you square -112, you must square the negative as well. This is because %28-112%29%5E2=-112%2A-112=12544.)



t+=+%28112+%2B-+sqrt%28+12544%2B-11520+%29%29%2F%282%2A16%29 Multiply -4%2A180%2A16 to get -11520



t+=+%28112+%2B-+sqrt%28+1024+%29%29%2F%282%2A16%29 Combine like terms in the radicand (everything under the square root)



t+=+%28112+%2B-+32%29%2F%282%2A16%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



t+=+%28112+%2B-+32%29%2F32 Multiply 2 and 16 to get 32

So now the expression breaks down into two parts

t+=+%28112+%2B+32%29%2F32 or t+=+%28112+-+32%29%2F32

Lets look at the first part:

x=%28112+%2B+32%29%2F32

t=144%2F32 Add the terms in the numerator
t=9%2F2 Divide

So one answer is
t=9%2F2



Now lets look at the second part:

x=%28112+-+32%29%2F32

t=80%2F32 Subtract the terms in the numerator
t=5%2F2 Divide

So another answer is
t=5%2F2

So our solutions are:
t=9%2F2 or t=5%2F2 (which are t=4.5 or t=2.5 respectively in decimal form)



So it takes 2.5 seconds for the arrow to climb to 180 ft and then 4.5 seconds for the arrow to fall back to 180 ft