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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

A rental: A rental car agency charges \(\$ 49.00\) per day and 25 cents per mile. a. Calculate the rental charge if you rent a car for 2 days and drive 100 miles. b. Use a formula to express the cost of renting a car as a function of the number of days you keep it and the number of miles you drive. Identify the function and each variable you use, and state the units. c. It is about 250 miles from Dallas to Austin. Use functional notation to express the cost to rent a car in Dallas, drive it to Austin, and return it in Dallas 1 week later. Use the formula from part b to calculate the cost.

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Renting motel rooms: You own a motel with 30 rooms and have a pricing structure that encourages rentals of rooms in groups. One room rents for \(\$ 85.00\), two for \(\$ 83.00\) each, and in general the group rate per room is found by taking \(\$ 2\) off the base of \(\$ 85\) for each extra room rented. a. How much money do you charge per room if a group rents 3 rooms? What is the total amount of money you take in? b. Use a formula to give the rate you charge for each room if you rent \(n\) rooms to an organization. c. Find a formula for a function \(R=R(n)\) that gives the total revenue from renting \(n\) rooms to a convention host. d. Use functional notation to show the total revenue from renting a block of 9 rooms to a group. Calculate the value.

Brightness of stars: The apparent magnitude \(m\) of a star is a measure of its apparent brightness as the star is viewed from Earth. Larger magnitudes correspond to dimmer stars, and magnitudes can be negative, indicating a very bright star. For example, the brightest star in the night sky is Sirius, which has an apparent magnitude of \(-1.45\). Stars with apparent magnitude greater than about 6 are not visible to the naked eye. The magnitude scale is not linear in that a star that is double the magnitude of another does not appear to be twice as dim. Rather, the relation goes as follows: If one star has an apparent magnitude of \(m_{1}\) and another has an apparent magnitude of \(m_{2}\), then the first star is \(t\) times as bright as the second, where \(t\) is given by $$ t=2.512^{m_{2}-m_{1}} . $$ The North Star, Polaris, has an apparent magnitude of \(2.04\). How much brighter than Polaris does Sirius appear?

A population of deer: When a breeding group of animals is introduced into a restricted area such as a wildlife reserve, the population can be expected to grow rapidly at first but to level out when the population grows to near the maximum that the environment can support. Such growth is known as logistic population growth, and ecologists sometimes use a formula to describe it. The number \(N\) of deer present at time \(t\) (measured in years since the herd was introduced) on a certain wildlife reserve has been determined by ecologists to be given by the function $$ N=\frac{12.36}{0.03+0.55^{t}} $$ a. How many deer were initially on the reserve? b. Calculate \(N(10)\) and explain the meaning of the number you have calculated. c. Express the number of deer present after 15 years using functional notation, and then calculate it. d. How much increase in the deer population do you expect from the 10 th to the 15 th year?
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