Page 53 - How To Solve Word Problems In Calculus
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EXAMPLE 15
The value of a machine after t years of service is
2
V(t) = 100t − 3000t + 20,000 dollars. At what rate does the
machine depreciate after 5 years?
Solution
The rate of depreciation is the rate at which value is lost.
If V(t) represents the value of the machine after t years, V (t)
represents the rate at which its value changes.
V (t) = 200t − 3000
V (5) =−2000
The machine depreciates at the rate of $2000 per year after
5 years.
Supplementary Problems
3
1. A closed boxwith a square base has a volume of 100 in . Compute
the rate of change of its surface area with respect to its base
dimension x when x = 5.
√
2. If f (x) = x, determine
(a) the rate of change of f with respect to x when x = 16
(b) the average rate of change of f from x = 16 to x = 25
3. The position of a moving object after t minutes of travel is
3
2
s(t) = 2t − 27t + 84t + 25 feet.
(a) What is the object’s average velocity during the first 2 minutes?
(b) Where is the object and how fast is it moving after 1 minute of
travel? 5 minutes? 10 minutes?
(c) What is the acceleration of the object after 1 minute? 5
minutes? 10 minutes?
(d) When is the velocity of the object 0? What is the object’s
position at that time?
(e) Describe the motion of the object during the first 10 minutes of
travel.
4. The height after t seconds of travel of a projectile thrown upward
2
from the top of a 96-ft building is h(t) = 80t − 16t + 96 ft.
40