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 arr

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Question 117872: 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.
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
You are given the equation for the height of the arrow above the ground as:
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h=+-16t%5E2%2B112t
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in which h represents the height above ground and t is the number of seconds after the arrow
leaves the ground.
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To find the value of t when the height is 180 feet, substitute 180 for h in the equation to get:
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180=+-16t%5E2%2B112t
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To get this equation into a little more standard form, begin by transposing (switching sides)
to get:
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-16t%5E2%2B112t+=+180
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Get rid of the 180 on the right side by subtracting 180 from both sides of the equation
to change the equation to:
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-16t%5E2+%2B+112t+-+180+=+0
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Make the first term positive by multiplying all terms on both sides of the equation by -1:
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16t%5E2+-+112t+%2B+180+=+0
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This equation can be simplified a little by dividing both sides (all terms) by 4 to reduce it to:
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4t%5E2+-+28t+%2B+45+=+0
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This can be solved by using the quadratic equation, but with a little work you can also
factor it into:
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%282t+-+5%29%2A%282t+-+9%29+=+0
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Notice that if either of the two factors on the left side is equal to zero, the multiplication
on the left side will involve a zero. So the entire left side will become zero and therefore
will equal the right side which is zero. So you can solve this equation by setting the two
factors equal to zero ... (one at a time).
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First set 2t - 5 equal to zero and solve for t:
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2t+-+5+=+0 <=== add 5 to both sides to get rid of the -5 on the left side
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2t+=+5 <=== solve for t by dividing both sides by 2 and you get
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t+=+5%2F2+=+2.5
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Next do the same process on the second factor. Set it equal to zero:
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2t+-+9+=+0 <=== add 9 to both sides to get rid of the -5 on the left side
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2t+=+9 <=== solve for t by dividing both sides by 2 and you get
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t+=+9%2F2+=+4.5
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What does this tell you? You have two answers ... t = 2.5 seconds and t = 4.5 seconds. These
are the two times when the arrow is 180 feet above the ground. How can that be?
...
Easy ... at 2.5 seconds the arrow is on its way up and is at 180 feet above ground. Later
at 4.5 seconds after launch the arrow is falling on its way down and is at 180 feet above the
ground.
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Hope this helps you to understand the problem and you can see how the solution is obtained.
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