A diver on a platform 40 feet in height jumps upward with an initial velocity of 5 feet per sec. His height in h feet after t seconds is given by the function h=-16t^2+5t+50. What is his maximum height? How long will it take him to reach the surface of the water?
There is a discrepancy in your problem.
You state the platform is 40 feet high, yet the equation suggests an initial height of 50 feet.
You would find the vertex of the downwards parabola.
An easy way is to find the x of the vertex by -b/2a
= -5/(-32) = 5/32 seconds
Now sub that back in to find the height when t = 5/32
for the last part,
set -16t^2 + 5t + 50 = 0
and solve for t using the quadratic formula.
I assume you have not covered calculus which would make this easy.
Without calculus you must complete the square to find the vertex of the parabola.
h=-16t^2+5t+50
-h = 16 t^2 - 5t - 50
50 - h = 16 t^2 - 5 t
50/16 -h/16 = t^2 -(5/16) t
50/16 - h/16 + 25/1024 = t^2 -5/16 t + 25/1024
3225/1024 - h/16 = (t-5/32)^2
we are at the top when t = 5/32
and h = 3225/64 = 50.4
Now what is t when h = 0?
= -16 t^2 +5 t + 50
t = [ -5 +/- sqrt (25 + 3200) ]/-32
t = [-5 + 56.8 ]/-32 = -1.62
ignore that solution, negative time was before we started
t = [-5 - 56.8]/-32 = 1.93 seconds
To find the maximum height of the diver, we need to determine the vertex of the quadratic function h(t) = -16t^2 + 5t + 50.
The vertex of a quadratic function in the form f(t) = at^2 + bt + c can be found using the formula t = -b / (2a).
In this case, a = -16 and b = 5.
Using the formula, we can calculate the time it takes to reach the maximum height:
t = -5 / (2 * -16)
t = -5 / -32
t = 5/32
So, it will take him 5/32 seconds to reach the maximum height.
Now, let's substitute this time back into the function h(t) to find the maximum height:
h(5/32) = -16(5/32)^2 + 5(5/32) + 50
h(5/32) = -16(25/1024) + 25/32 + 50
h(5/32) = -25/64 + 25/32 + 50
h(5/32) = -25/64 + 50/64 + 50
h(5/32) = 25/64 + 3200/64
h(5/32) = 3225/64
Therefore, the maximum height reached by the diver is 3225/64 feet.
To find the time it takes for the diver to reach the surface of the water, we set h(t) = 0 (since the surface of the water is at height 0) and solve for t.
0 = -16t^2 + 5t + 50
This quadratic equation can be solved by factoring or using the quadratic formula. For simplicity, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
t = (-5 ± √(5^2 - 4(-16)(50))) / (2(-16))
t = (-5 ± √(25 + 3200)) / (-32)
t = (-5 ± √(3225)) / (-32)
t = (-5 ± 5√(129)) / (-32)
Since the diver cannot have a negative time, we discard the negative value and take the positive value:
t = (-5 + 5√(129)) / (-32)
t ≈ 4.3 seconds
So, it will take him approximately 4.3 seconds to reach the surface of the water.