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Calculus / Calculus: Early Transcendentals 1 / Chapter 12.7 / Problem 40E

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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Travel cost The cost of a trip that is L miles long,

Chapter 10, Problem 40E

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QUESTION:

Travel cost The cost of a trip that is L miles long, driving a car that gets m miles per gallon, with gas costs of $p/gal is C = Lp/m dollars. Suppose you plan a trip of L = 1500 mi in a car that gets m = 32 mi/gal, with gas costs of p = $3.80/gal.

a. Explain how the cost function is derived.

b. Compute the partial derivatives \(C_{L}, C_{m}, \text { and } C_{p}\). Explain the meaning of the signs of the derivatives in the context of this problem.

c. Estimate the change in the total cost of the trip if L changes from L = 1500 to L = 1520, m changes from m = 32 to 31, and p changes from $3.80 to p = $3.85.

d. Is the total cost of the trip (with L = 1500 mi, m = 32 mi/gal, and p = $3.80) more sensitive to a 1% change in L, m, or p (assuming the other two variables are fixed)? Explain.

Questions & Answers

QUESTION:

Travel cost The cost of a trip that is L miles long, driving a car that gets m miles per gallon, with gas costs of $p/gal is C = Lp/m dollars. Suppose you plan a trip of L = 1500 mi in a car that gets m = 32 mi/gal, with gas costs of p = $3.80/gal.

a. Explain how the cost function is derived.

b. Compute the partial derivatives \(C_{L}, C_{m}, \text { and } C_{p}\). Explain the meaning of the signs of the derivatives in the context of this problem.

c. Estimate the change in the total cost of the trip if L changes from L = 1500 to L = 1520, m changes from m = 32 to 31, and p changes from $3.80 to p = $3.85.

d. Is the total cost of the trip (with L = 1500 mi, m = 32 mi/gal, and p = $3.80) more sensitive to a 1% change in L, m, or p (assuming the other two variables are fixed)? Explain.

ANSWER:

Solution 40E

Step 1

Consider the following problem

To derive the cost function

The total distance of trip is miles

And consumption of gas is mile per gallon

And cost of gas is

Since consumption of gas in mile distance is gallon

Therefore consumption of gas in 1 mile distance is

Since cost of 1 gallon gas is

Therefore cost of gallon gas is

Now cost of 1 mile distance is

Since total distance is miles

Therefore cost of distance is

Suppose total cost of trip is denoted by then

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