The formula h=300t- 5t^2 gives the height, h in meters, of a rocket the seconds after take off. The maximum height reached by the rocket is 4500m. How long will it take the rocket to reach this maximum height?
H=300t-5t^2=4500
60t-t^2-900=0
-t^2+60t-900=0
multiply by negative sign
t^2-60t+900=0
(t^2-30t)-(30t+900)=0
t(t-30)-30(t-30)=0
(t-30)^2=0
divide tru by (t-30)
you get t=30
To find the time it takes for the rocket to reach its maximum height, we need to determine the value of 't' when the height, h, is at its maximum.
Given that the height of the rocket, h, is given by the formula h = 300t - 5t^2, and the maximum height is 4500m, we can set up the equation:
4500 = 300t - 5t^2
Now, let's rearrange the equation to put it in standard quadratic form:
5t^2 - 300t + 4500 = 0
To find the value(s) of 't', we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 5, b = -300, and c = 4500. Let's substitute these values into the quadratic formula:
t = (300 ± √((-300)^2 - 4(5)(4500))) / (2(5))
Simplifying the formula further:
t = (300 ± √(90000 - 90000)) / 10
t = (300 ± √0) / 10
t = (300 ± 0) / 10
The discriminant (b^2 - 4ac) of the quadratic formula is zero, which means there is only one solution. Therefore, no matter which sign we choose in the ±, the resulting equation will be the same.
t = (300 + 0) / 10
t = 300 / 10
t = 30
Hence, the rocket will take 30 seconds to reach its maximum height of 4500m.