SOLUTION: Please help with the following word problem: A stone is thrown upward from a bridge. The stone's height in feet, "h", above the water "t" seconds after the stone is thrown is a fu

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Please help with the following word problem: A stone is thrown upward from a bridge. The stone's height in feet, "h", above the water "t" seconds after the stone is thrown is a fu      Log On


   

Question 149392: Please help with the following word problem:
A stone is thrown upward from a bridge. The stone's height in feet, "h", above the water "t" seconds after the stone is thrown is a function given by h=-16t2 + 32t + 256.
a. How tall is the bridge from which the stone was thrown? My answer is 256 ft.
b. How high above the water is the stone after 3 seconds? My answer is 208 ft.
c. What is the maximum height of the stone? Stuck on this question.
d. After how many seconds does the stone reach the maximum height? Stuck on this question too.
e. How long does it take for the stone to hit the water? My answer is 5.12 seconds.
Please help with questions c and d.
Any help would be greatly appreciated. Thank you.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a) correct
b) correct
c)

To find the max height, we need to find the vertex.




In order to find the vertex, we first need to find the x-coordinate of the vertex.


To find the x-coordinate of the vertex, use this formula: x=%28-b%29%2F%282a%29.


x=%28-b%29%2F%282a%29 Start with the given formula.


From y=-16t%5E2%2B32t%2B256, we can see that a=-16, b=32, and c=256.


x=%28-%2832%29%29%2F%282%28-16%29%29 Plug in a=-16 and b=32.


x=%28-32%29%2F%28-32%29 Multiply 2 and -16 to get -32.


x=1 Divide.


So the x-coordinate of the vertex is x=1. Note: this means that the axis of symmetry is also x=1.


Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.


y=-16t%5E2%2B32t%2B256 Start with the given equation.


y=-16t%5E2%2B32t%2B256 Plug in x=1.


y=-16t%5E2%2B32t%2B256 Start with the given equation.


y=-16%281%29%5E2%2B32%281%29%2B256 Plug in t=1.


y=-16%281%29%2B32%281%29%2B256 Square 1 to get 1.


y=-16%2B32%281%29%2B256 Multiply -16 and 1 to get -16.


y=-16%2B32%2B256 Multiply 32 and 1 to get 32.


y=272 Combine like terms.


So the y-coordinate of the vertex is y=272.


So the vertex is .


So the highest point on the graph of y=-16t%5E2%2B32t%2B256 is .


This means that maximum height of the stone is 272 feet




From the previous solution, the highest point is at the point . So when t=1, then h=272. So the stone reaches the highest point at 1 second




e) correct.