SOLUTION: The height h in feet of an object after t seconds is given by the function {{{h=-16t^2+90t+10}}} How long will it take the object to hit the ground?

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: The height h in feet of an object after t seconds is given by the function {{{h=-16t^2+90t+10}}} How long will it take the object to hit the ground?       Log On


   

Question 103994: The height h in feet of an object after t seconds is given by the function
h=-16t%5E2%2B90t%2B10
How long will it take the object to hit the ground?

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
The object will hit the ground when the height is 0, so let h=0

0=-16t%5E2%2B90t%2B10


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%2B90%2At%2B10=0 ( notice a=-16, b=90, and c=10)




t+=+%28-90+%2B-+sqrt%28+%2890%29%5E2-4%2A-16%2A10+%29%29%2F%282%2A-16%29 Plug in a=-16, b=90, and c=10



t+=+%28-90+%2B-+sqrt%28+8100-4%2A-16%2A10+%29%29%2F%282%2A-16%29 Square 90 to get 8100



t+=+%28-90+%2B-+sqrt%28+8100%2B640+%29%29%2F%282%2A-16%29 Multiply -4%2A10%2A-16 to get 640



t+=+%28-90+%2B-+sqrt%28+8740+%29%29%2F%282%2A-16%29 Combine like terms in the radicand (everything under the square root)



t+=+%28-90+%2B-+2%2Asqrt%282185%29%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+=+%28-90+%2B-+2%2Asqrt%282185%29%29%2F-32 Multiply 2 and -16 to get -32

So now the expression breaks down into two parts

t+=+%28-90+%2B+2%2Asqrt%282185%29%29%2F-32 or t+=+%28-90+-+2%2Asqrt%282185%29%29%2F-32


Now break up the fraction


t=-90%2F-32%2B2%2Asqrt%282185%29%2F-32 or t=-90%2F-32-2%2Asqrt%282185%29%2F-32


Simplify


t=45+%2F+16-sqrt%282185%29%2F16 or t=45+%2F+16%2Bsqrt%282185%29%2F16


So these expressions approximate to

t=-0.108998973130061 or t=5.73399897313006


So our possible solutions are:
t=-0.108998973130061 or t=5.73399897313006


However, since a negative time doesn't make sense, our only solution is t=5.73399897313006


So it takes about 5.7 seconds for the object to hit the ground.