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Part 1
A toy rocket is shot vertically into the air from a launching pad 4 feet above the ground with an initial velocity of 152 feet per second. The height h, in feet, of the rocket above the ground at t seconds after launch is given by the function h(t)=−16t2+152t+4. How long will it take the rocket to reach its maximum height? What is the maximum height?
To find the maximum height of the rocket, we first need to determine the time at which the rocket reaches its maximum height. We can do this by finding the value of t when the derivative of the height function, h(t), is equal to zero.
The derivative of h(t) = -16t^2 + 152t + 4 is given by:
h'(t) = -32t + 152
Setting h'(t) equal to zero:
-32t + 152 = 0
Solving for t, we get:
-32t = -152
t = 4.75
So, it will take the rocket 4.75 seconds to reach its maximum height.
To find the maximum height, we substitute the value of t into the height function:
h(4.75) = -16(4.75)^2 + 152(4.75) + 4
h(4.75) = -16(22.5625) + 722 + 4
h(4.75) = -361 + 722 + 4
h(4.75) = 365
Therefore, the maximum height of the rocket is 365 feet.
To find the time it takes for the rocket to reach its maximum height, we need to determine the time at which the rocket's velocity becomes zero. At the maximum height, the velocity of the rocket will be zero because it stops momentarily before starting to descend.
The velocity of the rocket at any time t is given by the derivative of the height function h(t) with respect to time:
v(t) = dh(t)/dt = -32t + 152
To find the time when the velocity is zero, we set v(t) equal to zero and solve for t:
-32t + 152 = 0
-32t = -152
t = -152 / -32
t = 4.75 seconds
So, it will take the rocket 4.75 seconds to reach its maximum height.
To find the maximum height, we substitute the value of t into the height function h(t):
h(4.75) = -16(4.75)^2 + 152(4.75) + 4
h(4.75) = -16(22.56) + 722 + 4
h(4.75) = -360.96 + 722 + 4
h(4.75) = 365.04
Therefore, the maximum height of the rocket is 365.04 feet.
To find the time it takes for the rocket to reach its maximum height, we need to determine when the rate of change of the height function, h(t), becomes zero. In other words, we need to find the value of "t" where the derivative of h(t) equals zero.
Step 1: Find the derivative of h(t) with respect to t.
h(t) = -16t^2 + 152t + 4
Taking the derivative of h(t) with respect to t, we get:
h'(t) = -32t + 152
Step 2: Set h'(t) equal to zero and solve for t.
-32t + 152 = 0
-32t = -152
t = -152 / -32
t = 4.75
Therefore, it will take the rocket approximately 4.75 seconds to reach its maximum height.
To find the maximum height, we substitute the value of t into the height function h(t).
h(4.75) = -16(4.75)^2 + 152(4.75) + 4
h(4.75) = -16(22.5625) + 722 + 4
h(4.75) = -360.5 + 722 + 4
h(4.75) = 365.5
Therefore, the maximum height reached by the rocket is approximately 365.5 feet.