If the annual rate of inflation averages over the next years, the approximate cost of goods or services during any year in that decade is where is the time in years and is the present cost. The price of an oil change for your car is presently . Find the rate of change of with respect to when . Round your answer to three decimal places.
A) 32.176 dollars per year
B) 1.966 dollars per year
C) 2.293 dollars per year
D) 5.962 dollars per year
E) 32.730 dollars per year

Question

If the annual rate of inflation averages  over the next  years, the approximate cost  of goods or services during any year in that decade is  where  is the time in years and  is the present cost. The price of an oil change for your car is presently  . Find the rate of change of  with respect to  when  . Round your answer to three decimal places.
A) 32.176 dollars per year 
B) 1.966 dollars per year 
C) 2.293 dollars per year 
D) 5.962 dollars per year 
E) 32.730 dollars per year

Quizplus · Accepted Answer

We can use the formula for the derivative of a power function:

f'(x) = kx^(k-1)

where k is a constant, and f(x) = kx^k

In this case, we have:

f(x) = C(1 + r)^t

where C is the current cost of an oil change, r is the annual rate of inflation, and t is the time in years. Taking the derivative with respect to t, we get:

f'(t) = Cr(1 + r)^(t-1)

Substituting the given values:

C = 49.95
r = 0.11595
t = 10

We get:

f'(10) = (49.95)(0.11595)(1.11595)^9 = 2.293

Rounding to three decimal places, we get:

f'(10) ≈ 2.293

Therefore, the answer is C.

If the Annual Rate of Inflation Averages Over the Next