SOLUTION: A toy rocket is launched from the top of a building 68 feet tall at an initial velocity of 182 feet per second. ​a) Give the function that describes the height of the ro

Algebra ->  Customizable Word Problem Solvers  -> Travel -> SOLUTION: A toy rocket is launched from the top of a building 68 feet tall at an initial velocity of 182 feet per second. ​a) Give the function that describes the height of the ro      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 1200687: A toy rocket is launched from the top of a building 68

feet tall at an initial velocity of 182

feet per second.
​a) Give the function that describes the height of the rocket in terms of time t.
​b) Determine the time at which the rocket reaches its maximum​ height, and the maximum height in feet.
​c) For what time interval will the rocket be more than 457

feet above ground​ level?
​d) After how many seconds will it hit the​ ground?
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
A toy rocket is launched from the top of a building 68 feet tall at an initial velocity of 182 feet per second.
​a) Give the function that describes the height of the rocket in terms of time t.
h(t) = h(t) = -16t^2 + 182t + 68
-------------
​b) Determine the time at which the rocket reaches its maximum​ height, and the maximum height in feet.
The max of the parabola is at t = -b/2a
t = -182/-32 = 5.6875 seconds
h(5.6875) = 585.5625 ft
----
​c) For what time interval will the rocket be more than 457 feet above ground​ level?
h(t) = -16t^2 + 182t + 68 = 457
16t^2 - 182t + 389 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 16x%5E2%2B-182x%2B389+=+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=%28-182%29%5E2-4%2A16%2A389=8228.

Discriminant d=8228 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--182%2B-sqrt%28+8228+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-182%29%2Bsqrt%28+8228+%29%29%2F2%5C16+=+8.52213511761214
x%5B2%5D+=+%28-%28-182%29-sqrt%28+8228+%29%29%2F2%5C16+=+2.85286488238786

Quadratic expression 16x%5E2%2B-182x%2B389 can be factored:
16x%5E2%2B-182x%2B389+=+%28x-8.52213511761214%29%2A%28x-2.85286488238786%29
Again, the answer is: 8.52213511761214, 2.85286488238786. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+16%2Ax%5E2%2B-182%2Ax%2B389+%29

The smaller value is ascending, the larger descending.
It's at or above between the 2 times.
--------------------
​d) After how many seconds will it hit the​ ground?
h(t) = -16t^2 + 182t + 68 = 0
Solve for t, ignore the negative value.