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 a

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Question 100530: 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 leaves the ground. Find the time it takes for the arrow to reach a height of 180 ft.
Found 2 solutions by stanbon, bucky:
Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website!
The equation h(t) = -16t^2 + 112t gives the height of an arrow, shot upward from te 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 to reach a height of 180ft.
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EQUATION:
-16t^2+112t= 180
-16t^2+112t-180 = 0
-4t^2+28t-45=0
t = [-28 +- sqrt(28^2-4*-4*-45)]/-8
t = [-28 +- sqrt(64)]/-8
t = [-28 +- 8]/-8
Positive answer:
t = [-36/-8] or t=[-20/-8]
t = 4.5 seconds or t=2.5 seconds
The 2.5 is the time on the way up; the 4.5 is the time on the way down.
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Cheers,
Stan H.
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Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website!
You are given the equation that you need to use to solve this problem. That equation is:
.

.
Since h represents the height in feet you can substitute that value into the equation to
get that:
.

.
From this point on, solving the equation for t is more just an exercise in mathematics
than it is a science problem.
.
The equation you now have is a quadratic equation. You need to get this into the standard
quadratic form of:
.

.
Do this by getting rid of the on the right side. And you can do that
by adding to both sides. This addition will cancel out the right side
(because the terms are equal but of opposite sign) and the equation then becomes:
.

.
You can reduce the size of the numbers involved by dividing both sides of this equation
(all terms) by 4 to get:
.

.
The most universal way of solving an equation such as this is to use the quadratic
formula which says that for a quadratic equation of the standard form:
.

.
the answers will be in the form:
.

.
If you compare your equation term by term with the standard form, you will see that a = 4,
b = -28, and c = +45. If you substitute these values into the answer form you find that
the answers will be:
.

.
Let's begin by working the terms under the radical. and .
Substitute these into the radical and the answer equation becomes:
.

.
The two terms under the radical combine to give:
.

.
and since the square root of 64 is 8 the equation further reduces to:
.

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Now you can simplify -(-28) by changing the sign of the term in the parentheses to get +28.
Further the denominator of (2*4) multiplies out to 8. Substituting these values into the
answer equation results in:
.

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So the two possible answers are:
.

.
and
.

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Now then, how do you interpret these answers? What they mean is that 2.5 seconds after launch
the arrow reaches 180 feet high on its way up, and 4.5 seconds after launch the arrow is
again at 180 feet on its way down.
.
Hope this helps you with the problem and shows you how you can use the quadratic formula to
solve quadratic equations.
.