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

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) (Show Source): You can put this solution on YOUR website!

Let h=180

Subtract 180 from both sides

Now let's use the quadratic formula to solve for t:

Starting with the general quadratic

the general solution using the quadratic equation is:

So lets solve ( notice , , and )

Plug in a=-16, b=112, and c=-180

Square 112 to get 12544

Multiply to get

Combine like terms in the radicand (everything under the square root)

Simplify the square root

Multiply 2 and -16 to get -32

So now the expression breaks down into two parts

or

Lets look at the first part:

Add the terms in the numerator
Divide

So one answer is

Now lets look at the second part:

Subtract the terms in the numerator
Divide

So another answer is

So our solutions are:
or (which are 2.5 and 4.5 in decimal form)

Notice when we graph (just replace t with x), we get:

and we can see that the roots are and . 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.
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