SOLUTION: I have a test tomorrow and i need to understand how to work complicated word problems on Quadratic functions. Here it is... Scouts used a slingshot to launch a small rock upward fr

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Question 27379: I have a test tomorrow and i need to understand how to work complicated word problems on Quadratic functions. Here it is... Scouts used a slingshot to launch a small rock upward from a cliff 250 feet above a river. The initial velocity was 320 feet per second. The formula h=250+320t-16t^2 gives the height in feet h of the rock relative to the river after t seconds. A) What is the maximum height reached by the rock? B) After how many seconds did the rock reach its maximum height? C) After how many seconds did the rock strike the river?
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
height=-16t^2+320t+250
This is a quadratic with a=-16, b=320, and c=250
a) The high point or "maximum" is at time t=-b/2a
t=-320/-32 = 10 seconds
Plug that time into the height formula to find the height
at t=10 seconds
h(10)=-16(10^2)+320(10)+250
= -160+320+250
= +160+250
= 410 feet
b) The rock will strike the ground when its height is zero.
So put "0" in place of height and solve for "t"
0=-16t^2+320t+250
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation at%5E2%2Bbt%2Bc=0 (in our case -16t%5E2%2B320t%2B250+=+0) has the following solutons:

t%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=%28320%29%5E2-4%2A-16%2A250=118400.

Discriminant d=118400 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-320%2B-sqrt%28+118400+%29%29%2F2%5Ca.

t%5B1%5D+=+%28-%28320%29%2Bsqrt%28+118400+%29%29%2F2%5C-16+=+-0.752906583803284
t%5B2%5D+=+%28-%28320%29-sqrt%28+118400+%29%29%2F2%5C-16+=+20.7529065838033

Quadratic expression -16t%5E2%2B320t%2B250 can be factored:
-16t%5E2%2B320t%2B250+=+-16%28t--0.752906583803284%29%2A%28t-20.7529065838033%29
Again, the answer is: -0.752906583803284, 20.7529065838033. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-16%2Ax%5E2%2B320%2Ax%2B250+%29

Cheers,
stan H.