Question
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.
(a)
Construct an exponential model for per capita whole milk consumption (in gallons) where years since 1970.
Answers
Reiny
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?
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?
Katie
Thanks for the help! It actually helped it... the answer was
27.4*(.961)^t))
27.4*(.961)^t))