h(t)= -16t^2 + 40t + 5 represents the height, in feet, of an object above level ground at time t, in seconds.

Question

Complete the following sentence with one of the choices listed below.

At t = 2 seconds, the object is ____________

A. rising at an increasing rate
B. rising at a decreasing rate
C. falling at an increasing rate
D. falling at a decreasing rate
E. not correctly described by any of the above

Answer 1

h'(t) = -32t + 40
h'(2) = -32(2) + 40
h'(2) = -64 + 40
h'(2) = -24 < 0 decreasing (or falling)
h''(t) = -32 < 0 concave down (the slope is getting steeper)
C) falling at an increasing rate

Answer 2

Well, at t = 0 it was at 5 feet.
At t = 2 it is at h = -64 + 80 + 5 = 21 feet (just checking to make sure it hasn't hit the ground already...if it had, the answer would be E regardless of the desire to differentiate the equation a couple of times!!)

If you differentiate the equation:

dh/dt = -32t + 40 = v
Evaluate at t = 2 seconds gives v = dh/dt = -24 feet/sec (ie - falling due to negative sign)

Differentiate again:
dv/dt = a = -32 feet/sec/sec (Earth gravitational acceleration)

It is accelerating in the same direction (sign) as its velocity at t = 2 seconds, therefore it is C (falling at an increasing rate).

Answer 3

h(t) = -16t^2 + 40t + 5

h(0) = -16(0)^2 + 40(0) + 5
h(0) = 5

h(1) = -16(1)^2 + 40(1) + 5
h(1) = -16(1) + 40 + 5
h(1) = -16 + 45
h(1) = 29

h(2) = -16(2)^2 + 40(2) + 5
h(2) = -16(4) + 80 + 5
h(2) = -64 + 85
h(2) = 21

as you can see from x = 0 to x = 1, it raised at an increasing rate, but from x = 1 to x = 2, it fell at an increasing rate

ANS : C. falling at an increasing rate

Answer 4

D. If you differentiate once, you find that its velocity is negative, so it's falling. If you differentiate again you find that its acceleration is negative, so it's slowing down. Correction C is the right answer. It's falling at a decreasing rate, like the previous answer says.

Answer 5

the acceleration is h'' or d/dt^2(h)= -32
This means that the acceleration is directed downward at a
constant rate.
the velocity is h' or d/dt(h)= -32t +40
Since t=2, the rate is directed downward at -24 f/s

The proper answer is C.

Answer 6

if there is a negative in front of the 16 the answer is C
if there is not a negative sign in front of the 16 the answer is A

Answer 7

C