SOLUTION: Solve the following problems by defining the variable, writing a quadratic equation, solving it an then providing a written conclusion. - A cannonball is fired form ground leve

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Question 247743: Solve the following problems by defining the variable, writing a quadratic equation, solving it an then providing a written conclusion.
- A cannonball is fired form ground level on an arc described by h = -t^2 + 28t where "h" is the height in meters at any time "t" in seconds. Determine the number of seconds it would take the ball to reach a height of 192m. At what other point in its flight is also at 192m?
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
A cannonball is fired form ground level on an arc described by h = -t^2 + 28t where "h" is the height in meters at any time "t" in seconds. Determine the number of seconds it would take the ball to reach a height of 192m. At what other point in its flight is also at 192m?
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h(t) = -t^2 + 28t = 192
t^2 - 28t + 192 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-28x%2B192+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-28%29%5E2-4%2A1%2A192=16.

Discriminant d=16 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--28%2B-sqrt%28+16+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-28%29%2Bsqrt%28+16+%29%29%2F2%5C1+=+16
x%5B2%5D+=+%28-%28-28%29-sqrt%28+16+%29%29%2F2%5C1+=+12

Quadratic expression 1x%5E2%2B-28x%2B192 can be factored:
1x%5E2%2B-28x%2B192+=+%28x-16%29%2A%28x-12%29
Again, the answer is: 16, 12. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-28%2Ax%2B192+%29

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It takes 12 seconds to reach 192 meters.
It's again at 192 meters on its way down at 16 seconds.
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The max height will be at 14 seconds, and the max is 196 meters.
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This must be on a small planet or asteroid to have a gravity of 2 m/sec/sec.
On Earth, it's 9.8 m/sec/sec, and the equation is -4.9t^2 for the acceleration term.