SOLUTION: The height of a ball thrown directly up from a roof 144 ft high with an initial velocity of 128 ft/sec is given by the formula y= 144 + 128t- 16t^2. What is the maximum height atta

Algebra ->  Trigonometry-basics -> SOLUTION: The height of a ball thrown directly up from a roof 144 ft high with an initial velocity of 128 ft/sec is given by the formula y= 144 + 128t- 16t^2. What is the maximum height atta      Log On


   

Question 215013: The height of a ball thrown directly up from a roof 144 ft high with an initial velocity of 128 ft/sec is given by the formula y= 144 + 128t- 16t^2. What is the maximum height attained by the ball?

At first, the problem looked to be a quadratic equation... but then I assumed it had something to do with the velocity formula. Essentially, I am a bit lost.
Thanks in advance!
-B
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Note: I'm going to use the variable 'x' instead of 't'.


y=+144+%2B+128x-+16x%5E2 Start with the given equation.


y=-16x%5E2%2B128x%2B144 Rearrange the terms.


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=-16x%5E2%2B128x%2B144, we can see that a=-16, b=128, and c=144.


x=%28-%28128%29%29%2F%282%28-16%29%29 Plug in a=-16 and b=128.


x=%28-128%29%2F%28-32%29 Multiply 2 and -16 to get -32.


x=4 Divide.


So the x-coordinate of the vertex is x=4. Note: this means that the axis of symmetry is also x=4.


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


y=-16x%5E2%2B128x%2B144 Start with the given equation.


y=-16%284%29%5E2%2B128%284%29%2B144 Plug in x=4.


y=-16%2816%29%2B128%284%29%2B144 Square 4 to get 16.


y=-256%2B128%284%29%2B144 Multiply -16 and 16 to get -256.


y=-256%2B512%2B144 Multiply 128 and 4 to get 512.


y=400 Combine like terms.


So the y-coordinate of the vertex is y=400.


So the vertex is .


This means that the max height is 400 feet (since the max/min is the y coordinate of the vertex). This max height occurs when t=4 (since x=4 at the vertex and I used 'x' instead of 't')