SOLUTION: 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 leaves the ground

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: 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 leaves the ground      Log On


   

Question 89021: 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 leaves the ground. Find the time it takes for the arrow toreach a height of 180 ft.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
h+=+-16t%5E2+%2B+112t+

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

0+=+-16t%5E2+%2B+112t-180+ Subtract 180 from both sides


Now 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%2B112%2At-180=0 ( notice a=-16, b=112, and c=-180)

t+=+%28-112+%2B-+sqrt%28+%28112%29%5E2-4%2A-16%2A-180+%29%29%2F%282%2A-16%29 Plug in a=-16, b=112, and c=-180



t+=+%28-112+%2B-+sqrt%28+12544-4%2A-16%2A-180+%29%29%2F%282%2A-16%29 Square 112 to get 12544



t+=+%28-112+%2B-+sqrt%28+12544%2B-11520+%29%29%2F%282%2A-16%29 Multiply -4%2A-180%2A-16 to get -11520



t+=+%28-112+%2B-+sqrt%28+1024+%29%29%2F%282%2A-16%29 Combine like terms in the radicand (everything under the square root)



t+=+%28-112+%2B-+32%29%2F%282%2A-16%29 Simplify the square root



t+=+%28-112+%2B-+32%29%2F-32 Multiply 2 and -16 to get -32

So now the expression breaks down into two parts

t+=+%28-112+%2B+32%29%2F-32 or t+=+%28-112+-+32%29%2F-32

Lets look at the first part:

x=%28-112+%2B+32%29%2F-32

t=-80%2F-32 Add the terms in the numerator
t=5%2F2 Divide

So one answer is
t=5%2F2



Now lets look at the second part:

x=%28-112+-+32%29%2F-32

t=-144%2F-32 Subtract the terms in the numerator
t=9%2F2 Divide

So another answer is
t=9%2F2

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

Notice when we graph -16%2Ax%5E2%2B112%2Ax-180 (just replace t with x), we get:

+graph%28+500%2C+500%2C+-5%2C+19%2C+-5%2C+19%2C-16%2Ax%5E2%2B112%2Ax%2B-180%29+

and we can see that the roots are t=5%2F2 and t=9%2F2. This verifies our answer


So it takes 2.5 seconds to rise to the height of 180 feet, and 4.5 seconds to hit 180 feet.