SOLUTION: For 1989 and 1990 Dave Johnson had the highest decathlon score in the world. When Johnson reached a speed of 32 ft/sec on the pole vault runway, his height above the ground t sec

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Question 147162: For 1989 and 1990 Dave Johnson had the highest decathlon score in the world. When Johnson reached a speed of 32 ft/sec on the pole vault runway, his height above the ground t seconds after leaving the ground was given
by h= -16t^2+32t . (The elasticity of the pole converts the horizontal speed into vertical speed.) Determine how long Johnson was in the air. For how long was he more than 14 ft in the air?
Answer by nerdybill(5411) About Me  (Show Source):
You can put this solution on YOUR website!
h= -16t^2+32t
The problem asks how long he was ABOVE 14 feet -- which is h (height). Simply plug it into the equation and solve for t (time).
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h= -16t^2+32t
14 = -16t^2+32t
Move everything to the left side:
16t^2 - 32t + 14 = 0
Divide by throughout by 2:
8t^2 - 16t + 7 = 0
.
Since you can't factor, you must use the quadratic equation:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 8x%5E2%2B-16x%2B7+=+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=%28-16%29%5E2-4%2A8%2A7=32.

Discriminant d=32 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--16%2B-sqrt%28+32+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-16%29%2Bsqrt%28+32+%29%29%2F2%5C8+=+1.35355339059327
x%5B2%5D+=+%28-%28-16%29-sqrt%28+32+%29%29%2F2%5C8+=+0.646446609406726

Quadratic expression 8x%5E2%2B-16x%2B7 can be factored:
8x%5E2%2B-16x%2B7+=+8%28x-1.35355339059327%29%2A%28x-0.646446609406726%29
Again, the answer is: 1.35355339059327, 0.646446609406726. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+8%2Ax%5E2%2B-16%2Ax%2B7+%29

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x has two solutions:
1.35355339059327, 0.646446609406726
The difference is the time he was above 14 feet:
1.35355339059327 - 0.646446609406726 = .0607 seconds