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Chapter 1: Problem 6

A car that gets 32 miles per gallon: The cost \(C\) of operating a certain car that gets 32 miles per gallon is a function of the price \(g\), in dollars per gallon, of gasoline and the distance \(d\), in miles, that you drive. The formula for \(C=C(g, d)\) is \(C=g d / 32\) dollars. a. Use functional notation to express the cost of operation if gasoline costs 98 cents per gallon and you drive 230 miles. Calculate the cost. b. Calculate \(C(1.03,172)\) and explain the meaning of the number you have calculated.

Short Answer

Expert verified
a. \$7.04; b. \$5.54 for 172 miles at \$1.03/gal.

Step by step solution

01

Understand the Formula

The given formula for cost is \(C(g, d) = \frac{gd}{32}\), where \(g\) is the price per gallon and \(d\) is the distance in miles. This formula calculates the total cost by dividing the product of the price and distance by the car's 32 miles per gallon efficiency.
02

Convert Given Units for Part a

Gasoline costs 98 cents per gallon, which is \$0.98 per gallon since 98 cents equals 0.98 dollars. The distance \(d\) is 230 miles.
03

Substitute Values in Formula for Part a

Substitute \(g = 0.98\) dollars/gallon and \(d = 230\) miles into the formula: \[ C(0.98, 230) = \frac{0.98 \times 230}{32} \]
04

Calculate the Cost for Part a

Calculate the expression: \(0.98 \times 230 = 225.4\). Now divide by 32: \(\frac{225.4}{32} \approx 7.04375\). Hence, the cost is approximately \$7.04.
05

Define Inputs for Part b

For part b, the price of gasoline is \$1.03 per gallon and the distance is 172 miles. These values are given as \(g = 1.03\) dollars/gallon and \(d = 172\) miles.
06

Substitute Values in Formula for Part b

Substitute \(g = 1.03\) and \(d = 172\) into the formula: \[ C(1.03, 172) = \frac{1.03 \times 172}{32} \]
07

Calculate the Cost for Part b

Calculate the expression: \(1.03 \times 172 = 177.16\). Now divide by 32: \(\frac{177.16}{32} \approx 5.53625\). Hence, the cost is approximately \$5.54.
08

Interpret Result for Part b

The calculated result \(C(1.03, 172) \approx 5.54\) signifies that it costs approximately \\(5.54 to drive 172 miles when the price of gasoline is \\)1.03 per gallon.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cost Calculation
When we talk about cost calculation in relation to cars, it's all about determining how much you spend on fuel while covering a certain distance. Here, we use the formula \(C(g, d) = \frac{gd}{32}\) to assess the expense. This accounts for the price of gasoline, \(g\), and the distance driven, \(d\). By inputting these two values into the formula, you can compute the overall fuel cost.

Let's break that down further:
  • Multiply the price per gallon by the miles driven to get the total expenditure on fuel for the entire journey.
  • Then divide by the car's miles per gallon efficiency, which is 32 in this case.
This operation gives you the cost of operating the car over the specified distance, providing a clear way to budget your driving expenses based on fluctuating gasoline prices.
Miles Per Gallon
Miles per gallon, often abbreviated as MPG, is a crucial efficiency metric reflecting how far a car can travel on one gallon of fuel. In our scenario, the car has a fuel efficiency of 32 miles per gallon, which is a moderately efficient number for a vehicle.

Consider these points about MPG:
  • If a car can travel more miles per gallon, it is considered more fuel-efficient. This generally means you spend less on gasoline for the same distance compared to less efficient cars.
  • MPG is an essential factor for environmentally-conscious consumers, as higher MPG generally correlates with lower emissions.
Hence, understanding and calculating MPG helps in making economically and environmentally sound decisions related to vehicle selection and fuel budgeting.
Functional Relationship
In mathematics, a functional relationship describes how one quantity depends on one or more other quantities. Here, the cost \(C\) depends functionally on both the gasoline price \(g\) and the distance \(d\). This relationship is expressed in the function notation as \(C(g, d)\), indicating that \(C\) is a function of \(g\) and \(d\).

This means:
  • The cost is directly proportional to the gasoline price; increasing \(g\) raises \(C\) if all else remains constant.
  • The cost is also directly proportional to the distance driven; more miles increase \(C\).
Understanding this relationship aids in predicting how changes in gasoline prices and driving habits impact overall fuel costs, which is vital for planning your travel and budgeting.
Algebraic Formula
An algebraic formula is a mathematical expression that calculates or determines a certain value. In this exercise, the formula \(C(g, d) = \frac{gd}{32}\) is used to determine the cost of operating the vehicle. This formula is derived from the equation of a line and uses arithmetic operations to produce a result based on inputs.

To effectively use an algebraic formula:
  • Substitute the known values into the formula, such as the current gasoline price and miles to be driven.
  • Perform the arithmetic operations as indicated鈥攎ultiplication first, followed by division.
This specific formula helps to efficiently calculate the operational costs based on varying gasoline prices and driving distances, making it a powerful tool for financial planning related to vehicle use.

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Most popular questions from this chapter

Loan origination fee: Lending institutions often charge fees for mortgages. One type of fee consists of closing costs. Closing costs often comprise a fixed fee for a confusing list of necessities associated with the loan. Additionally, some mortgages require a loan origination fee or points, which is a fixed percentage of the mortgage. At a certain institution the closing costs are \(\$ 2500\) and the points are \(2 \%\) of the mortgage amount. a. What are the fees for securing a mortgage of \(\$ 322,000\) ? b. Use a formula to express the loan fees \(F\), in dollars, associated with a mortgage of \(M\) dollars.

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Preparing a letter: You pay your secretary \(\$ 9.25\) per hour. A stamped envelope costs 50 cents, and paper costs 3 cents per page. a. How much does it cost to prepare and mail a 3 -page letter if your secretary spends 2 hours on typing and corrections? b. Use a formula to express the cost of preparing and mailing a letter as a function of the number of pages in the letter and the time it takes your secretary to type it. Identify the function and each of the variables you use, and state the units. c. Use the function you made in part b to find the cost of preparing and mailing a 2 -page letter that it takes your secretary 25 minutes to type. (Note: 25 minutes is \(\frac{25}{60}\) hour.)

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