The yearly per capita consumption of whole milk in the United States reached a peak of 40 gallons in 1945, at the end of World War II. By 1970 consumption was only 27.4 gallons per person. It has been steadily decreasing since 1970 at a rate of about 3.9% per year.

(a)
Construct an exponential model for per capita whole milk consumption (in gallons) where years since 1970.

Thanks for the help! It actually helped it... the answer was

27.4*(.961)^t))

assume an equation of the kind,

C = a e^(kt), where C is the consumption, a is the intital amount , t is the number of years since 1945, and k is a constant.

given:
when t = 0, C = 40,
40 = a e(k(0))
a = 40
so we have:
C = 40 e^(kt)

given : when t = 25 (1970), C = 27.4
27.4 = 40 e(25k)
take ln of both sides
ln 27.4 = ln 40 + 25k
k = (ln 27.4 - ln 40)/25
= -.015133..

C = 40 e^(-.015133.. t) <------ equation #1

or with a base of .961
C = 40 (.961)^(kt)
when t = 25, C = 27.4

27.4 = 40 (.961)^(25k)
log both sides:
log 27.4 = log 40 + 25k(log .961)
(log 27.4 - log 40)/(25log.961) = k
k = .38042

C = 40 (.961)^(.38042t) <----- second version

checking the last one:
if t = 25
C = 40 (.961^9.5105)
= 27.400001306 , pretty good eh?

To construct an exponential model for per capita whole milk consumption (in gallons), we can use the formula:

P(t) = P0 * e^(kt)

Where:
P(t) is the per capita whole milk consumption at time t
P0 is the initial per capita whole milk consumption (in 1970)
e is the base of the natural logarithm (approximately 2.71828)
k is the growth rate as a decimal

Given that the per capita whole milk consumption in 1970 was 27.4 gallons and it has been steadily decreasing at a rate of about 3.9% per year, we can calculate the growth rate as follows:

k = -0.039

Now we can plug in the values into the exponential model:

P(t) = 27.4 * e^(-0.039t)

This is the exponential model for per capita whole milk consumption in the United States, where t represents the number of years since 1970.

To construct an exponential model for per capita whole milk consumption, we need to find the general form of an exponential function. An exponential function is of the form:

y = ab^x

where y is the output variable, a is the initial value of the function, b is the growth factor or rate, and x is the input variable.

In this case, the initial value in 1970 is 27.4 gallons per person. We are given that the consumption decreases at a rate of about 3.9% per year. This means that the growth factor b is 1 - (3.9/100) = 0.961.

Using this information, we can construct the exponential model:

y = 27.4 * (0.961)^x

where x represents the number of years since 1970.

Therefore, the exponential model for per capita whole milk consumption is:

Per capita whole milk consumption (in gallons) = 27.4 * (0.961)^x