Question

Two entertainment venues are planning to expand the size of their audiences and each has decided on the rate at which it can increase its audience size. The functions below represent the prospective audience sizes of the venues after t years.

City Arts Center : C(t) = 250(1.45)^t/5
Oakwood Auditorium : C(t) = 200(1.55)^t/7

Which statement is true about the audience size of the venues?

The annual rate of increase in the audience size of Oakwood Auditorium is greater than the annual rate of increase in the audience size of City Arts Center.

The annual rate of increase in the audience size of Oakwood Auditorium is less than the annual rate of increase in the audience size of City Arts Center.

The annual rate of increase in the audience size of Oakwood Auditorium is the same as the annual rate of increase in the audience size of City Arts Center.

The annual percentage rates of increase in the audience sizes cannot be determined from this information.

Answer 1

Answer: The annual rate of increase in the audience size of Oakwood Auditorium is less than the annual rate of increase in the audience size of City Arts Center.

Explanation:

Here, the function that shows the audience sizes of the venues after t years. by City Arts Center is,

`C(t) = 250(1.45)^{(t)/(5)}`

⇒
`C(t) = 250((1.45)^{(1)/(5)})^t\implies =250(1.07714358779)^t`

Thus, the increasing rate of the function is 0.07714358779.

Similarly, the function that shows the audience sizes of the venues after t years. by Oakwood Auditorium is,

`C(t) = 200((1.55)^(1)/(7))^t\implies C(t) = 200(1.06460926794)^t`

Thus, the increasing rate of the function is 0.06460926794.

Since, 0.07714358779 > 0.06460926794

Hence, The annual rate of increase in the audience size of Oakwood Auditorium is less than the annual rate of increase in the audience size of City Arts Center.

Answer 2

Let us see... ideally we would like to have all equations with the same exponent or the same base so that we can compare the rates. Since the unknown is in the exponent, we have to work with them. In general,
`x^(y/z)= \sqrt[z]{x^y}`.
Applying this to the exponential parts of the functions, we have that the first equation is equal to:
250*(
`\sqrt[5]{1.45} ^t`)=250*(1.077)^t
The second equation is equal to: 200* (1.064)^t in a similar way.
We have that the base of the first equation is higher, thus the rate of growth is faster in the first case; Choice B is correct.