Choose all that are best modeled by an exponential function. A town with an initial population of 2,000 doubling in size every 10 years. A car rental that cost $20 a day plus $0.05 per mile. A bank account with a starting balance of $1,000 compounded annually at 5%. A sugar processing plant producing 2 tons of sugar per month.

Exponential functions for the following example will be given as follows:

a] A town with an initial population of 2,000 doubling in size every 10 years.

solution:

The exponential function for this example will be:

f (t) = ab^ (nt)

where:

a=initial size

b=growth rate

t=time

n=number of terms

thus plugging in our values we obtain:

f (t) = 2000 (2) ^t/10

b] A car rental that cost $20 a day plus $0.05 per mile.

This example will follow a linear equation of the form y=ax+b, where a=rate,

b=constant amount.

From our example:

a=$0.05 per mile

b=20 dollars per day

x=number of miles

Thus the function will be:

f (x) = 0.05x+20

c] A bank account with a starting balance of $1,000 compounded annually at 5%.

Here we shall use the compound interest formula to obtain our value:

FV=P (1+r/100) ^t

where:

FV=future value

P=principle amount

r=rate

t=time

thus, plugging in our value in the equation we obtain:

P=$1000; r=5%, time=t

f (t) = 1000 (1+5/100) ^t

f (t) = 1000 (1.05) ^t

Hence the function representing the case is:

f (t) = 1000 (1.05) ^t

d] A sugar processing plant producing 2 tons of sugar per month.

This will be modeled using linear function, such that the total will be given by:

f (x) = 2x

where:

x is the number of months:

thus the cases that are best modeled by exponential function is:

A and C