Can someone check this for me?

Suppose that the number of cars, C, on 1st Avenue in a city over a period of time t, in months, is graphed on a rectangular coordinate system where time is on the horizontal axis. Suppose that the number of cars driven on 1st Avenue can be modeled by an exponential function, C= p * at where p is the number of cars on the road on the first day recorded. If you commuted to work each day along 1st Avenue, would you prefer that the value of be between 0 and 1 or larger than 1? Explain your reasoning.

C = p *a^t

C = ?

P = 10 cars

T = 10 months

A = 0

C = 15(0)^10
C = 15(0)
C = 0

If a is greater than one, the number of cars on the road every day will get bigger and bigger, say a = 1.5

day 1 p*1.5
day 2 p*2.25

If a is smaller than one, c will get smaller every day after start, say a = .5
day 1 .p*.5
day 2 P*.25
day 3 p*.125

To answer the question, we need to understand the relationship between the value of "a" in the exponential function and the number of cars on 1st Avenue over time.

In the given exponential function, C = p * a^t, "a" is the base of the exponential function. If "a" is a value between 0 and 1, it means that the base is less than 1. When the base is less than 1, the exponential function will have a decay pattern. This means that as time increases, the value of the exponential function decreases.

On the other hand, if "a" is larger than 1, it means that the base is greater than 1. In this case, the exponential function will have a growth pattern. As time increases, the value of the exponential function increases.

Now, let's apply this understanding to the situation. If you commute to work each day along 1st Avenue, it would be preferable to have the value of "a" between 0 and 1.

Here's why:

- If "a" is between 0 and 1, it means that the base of the exponential function is less than 1. This implies that the number of cars on 1st Avenue will gradually decrease over time. As a commuter, this would mean less traffic congestion on the road, making your commute smoother and faster.

- On the other hand, if "a" is larger than 1, it means that the base of the exponential function is greater than 1. This would result in an exponential growth of the number of cars on 1st Avenue over time. As a commuter, this would lead to more traffic congestion, making your commute slower and less convenient.

Therefore, it is preferable for the value of "a" to be between 0 and 1 if you commute to work each day along 1st Avenue.