Question

an object is launched with a velocity at 64 feet per second from a platform 30 feet high. the function H(t)= -16t2+64t+30 represents the function where T is time in seconds, and H is height in feet.

what is the maximum height of the object ever reaches?
a) 92 ft
b) 94 ft
c) 2 ft
d) 94 ft

when does it reach the maximum height?

a) 2 sec
b) 94 sec
c) -2 sec
d) 3 sec

Answer 1

To find the maximum or minimum value of a function, we can find the derivative of the function, set it equal to 0, and solve for the critical points.

H'(t) = -32t + 64

Now find the critical numbers:

-32t + 64 = 0

-32t = -64

t = 2 seconds

Since H(t) has a negative leading coefficient, we know that it opens downward. This means that the critical point is a maximum value rather than a minimum. If we weren't sure, we could check by plugging in a value for t slightly less and slighter greater than t=2 into H'(t):

H'(1) = 32

H'(3) = -32

As you can see, the rate of change of the object's height goes from increasing to decreasing, meaning the critical point at t=2 is a maximum.

To find the height, plug t=2 into H(t):

H(2) = -16(2)^2 +64(2) + 30 = 94

The answer is 94 ft at 2 sec.