The amount spent by a consumer on nondurable goods can be modeled as n ( x ) = 1

Question

The amount spent by a consumer on nondurable goods can be modeled as

n(x) = 1.1 + 1.64 ln x thousand dollars

and the amount spent on motor vehicles can be modeled as

m(x) = 1.62(1.26x) hundred dollars

where x thousand dollars is the amount spent by that same consumer for all personal consumption.† (Note: Use the concept of inverse functions or models.)

(a) Use output values corresponding to $4,500 and $11,000 personal consumption to determine the appropriate function for the model that gives the amount spent by the consumer for all personal consumption in thousand dollars, where m is the amount spent on motor vehicles. (Round all numerical values to three decimal places.)
x(m) = ________ thousand dollars

(b) Write the function for the model that gives the amount spent on nondurable goods in thousand dollars, where m is the amount spent on motor vehicles. (Round all numerical values to three decimal places.)
n(x(m)) = _______ thousand dollars

(c) How much is spent on nondurable goods by somebody who spends $380 on his or her motor vehicle? (Round your answer to three decimal places.)
$ ________ thousand

At what rate is this amount changing with respect to motor vehicle spending? (Round your answer to three decimal places.)
$_________ thousand per hundred dollars

Write a sentence of interpretation for the results.

A consumer who spends $380 on his or her motor vehicles will also spend $ ________ thousand on nondurable goods. At the $380 level of spending on motor vehicles, the amount spent on nondurable goods is increasing  by $ __________ thousand per hundred dollars increase in spending on motor vehicles.

Explanation / Answer

(a)

For x = 4500, x(m) = 1.621.26*4.5 = 15.415

For x = 11000, x(m) = 1.621.26*11 = 801.453

(b)

For x(m) = 15.415, n(x(m)) = -1.1+1.64ln(15.415) = 2.770

For x(m) = 801.453, n(x(m)) = -1.1+1.64ln(801.45) = 26.224

(c)

For x(m) = 380, n(x(m)) = -1.1+1.64ln(380) = 17.789

(d)

Rate of change = (26.224-17.789)/(801.45-38)*100 = 1.105 thousand per hundred dollars

A consumer who spends $380 on his or her motor vehicles will also spend $ __17.789____ thousand on nondurable goods. At the $380 level of spending on motor vehicles, the amount spent on nondurable goods is increasing  by $ ___1.105____ thousand per hundred dollars increase in spending on motor vehicles.

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