SOLUTION: A fastball is hit straight up over home plate. The ball's height, h (in feet), from the ground is modeled by h(t)=-16t^2+80t+5, where t is measured in seconds. (a) Write an equat

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Question 721753: A fastball is hit straight up over home plate. The ball's height, h (in feet), from the ground is modeled by h(t)=-16t^2+80t+5, where t is measured in seconds.
(a) Write an equation to determine how long it will take for the ball to reach the ground. Solve the equation using the quadtratic formula.
(b) What is the maximum height of the ball above the ground?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
a) Since the ground would have a height of zero, the equation you would use is:
0=-16t%5E2%2B80t%2B5
As instructed in the problem, use the quadratic formula to find t. I'll get you started:
t+=+%28-%2880%29%2B-sqrt%28%2880%29%5E2-4%28-16%29%285%29%29%29%2F2%28-16%29
Now simplify. You will get two values for t. But one of them will be negative. Since negative time makes no sense, reject the negative solution.

b) h%28t%29=-16t%5E2%2B80t%2B5
We should know a little about what the graph of this equation would look like. Because of the t%5E2 this will be a parabola. Because the t%5E2 has a negative coefficient, -16, it will be a parabola that opens downward. If you picture (or draw) such a parabola, you will notice that the highest point on the graph is the vertex of the parabola. So to find the highest point, we need to find the vertex of the parabola. There are at least a couple of ways to do this:
  • Transform the equation into vertex form: %28t-h%29%5E2+=+4p%28y-k%29. This involves what is called "completing the square" which is a bit tedious.
  • Memorize and use the fact that the x (or in this case t) coordinate of the vertex will always be -b/2a. Then, once you have the x (or in this case t) coordinate of the vertex you can use that value and the equation to find the maximum value (or minimum value) of the function. (If the parabola opens upward the vertex is the minimum value.)
Since the second way is faster if you remember the formula, I'm going to use it. The t coordinate of the vertex will be:
-80%2F2%28-16%29+=+80%2F32+=+5%2F2
Now we can use this and the equation to find the maximum height:
h%285%2F2%29+=+-16%285%2F2%29%5E2%2B80%285%2F2%29%2B5
h%285%2F2%29+=+-16%2825%2F4%29%2B80%285%2F2%29%2B5
h%285%2F2%29+=+-100%2B200%2B5
h%285%2F2%29+=+105
So the maximum height of the ball is 105 feet.