Question

Jordan is examining her car expenses. Her car uses gasoline at a rate of \(6 \mathrm{L} / 100 \mathrm{km}\) and the average cost of a litre of gasoline where she lives is \(1.23\) a) Write the function, \(g(d),\) that relates the distance, \(d,\) in kilometres, driven to the quantity, \(g\), in litres, of gasoline used. b) Write the function, \(c(g),\) that relates the quantity, \(g\), in litres, of gasoline used to the average cost, \(c,\) in dollars, of a litre of gasoline. c) Write the composite function that expresses the cost of gasoline in terms of the distance driven. How much would it cost Jordan to drive \(200 \mathrm{km}\) in her car? d) Write the composite function that expresses the distance driven in terms of the cost of gasoline. How far could Jordan drive her car on S40?

Short answer

a) \( g(d) = \frac{6}{100}d \); b) \( c(g) = 1.23g \); c) The cost for 200 km is \$14.76; d) Jordan can drive approximately 542.68 km on \$40.

Step-by-step solution

Define the function for gasoline usage

The rate of gasoline usage is given as 6 liters per 100 kilometers. The function that relates the distance traveled, \(d\), to the volume of gasoline used, \(g\), can be written as:\[ g(d) = \frac{6}{100}d \]

Define the function for cost based on gasoline

The cost of gasoline is given as $1.23 per litre. The function that relates the volume of gasoline used, \(g\), to the total cost, \(c\), can be written as:\[ c(g) = 1.23g \]

Create composite function for cost based on distance

To find the cost as a function of distance driven, we need to combine the two functions. Substitute \(g(d)\) into \(c(g)\):\[ c(d) = c(g(d)) = c\bigg(\frac{6}{100}d\bigg) = 1.23 \times \frac{6}{100}d = 0.0738d \]Now, substitute \(d = 200\), to find the cost for driving 200 km:\[ c(200) = 0.0738 \times 200 = 14.76 \]Therefore, it would cost \$14.76 to drive 200 km.

Create composite function for distance based on cost

To express the distance driven as a function of the cost, we need to find the inverse of the composite function from Step 3. Starting from:\[ c(d) = 0.0738d \]Solve for \(d\):\[ d = \frac{c}{0.0738} \]Next, substitute \(c = 40\), to find the distance Jordan can drive with \$40:\[ d = \frac{40}{0.0738} \ d \text{ is approximately } 542.68 \text{ km} \]Therefore, Jordan can drive approximately 542.68 km on \$40.

Key concepts

Distance and Fuel Consumption

Understanding how distance and fuel consumption are related is essential for calculating car expenses. Jordan's car uses gasoline at a rate of 6 liters for every 100 kilometers driven. We can convert this relationship into a function. If Jordan drives a distance of \(d\) kilometers, the amount of gasoline \(g\) her car will use can be calculated by the function:
\[ g(d) = \frac{6}{100}d \]
This function means that for every kilometer driven, the car consumes \(0.06\) liters of gasoline. Thus, knowing the distance Jordan plans to drive helps us determine the fuel consumption easily.

Cost Function

After determining how much gasoline Jordan's car uses, we need to calculate the cost. The cost function relates the amount of gasoline used to the total cost. Here, the cost of gasoline is \(1.23\) dollars per liter. We can write a function for the cost based on the amount of gasoline \(g\). The function is:
\[ c(g) = 1.23g \]
By using this function, we can convert the volume of gasoline consumed into a monetary value, helping Jordan understand how much she spends on gasoline for any trip.

Composite Functions

Composite functions combine two or more functions to provide a new relationship between variables. In this context, we want to combine Jordan's fuel consumption function \(g(d)\) with her cost function \(c(g)\) to find the total cost based on the distance driven. We substitute \(g(d)\) into \(c(g)\):
\[ c(d) = c(g(d)) = c\bigg(\frac{6}{100}d\bigg) = 1.23 \times \frac{6}{100}d = 0.0738d \]
This composite function \(c(d) = 0.0738d\) now directly relates the distance driven to the total cost. For example, if Jordan wants to know the cost of driving 200 kilometers:
\[ c(200) = 0.0738 \times 200 = 14.76 \]
Therefore, it will cost Jordan \$14.76 to drive 200 kilometers.

Function Inversion

Function inversion helps us reverse the process, determining one variable when we know the other. To express the distance \(d\) driven based on the total cost \(c\), we need to invert the composite cost function. Starting from:
\[ c(d) = 0.0738d \]
Solving for \(d\) gives:
\[ d = \frac{c}{0.0738} \]
This inversion allows us to calculate how far Jordan can drive given a specific budget. For example, if Jordan has \$40 to spend on gasoline:
\[ d = \frac{40}{0.0738} \approx 542.68 \text{ km} \]
Hence, Jordan can drive approximately 542.68 kilometers with \$40, giving her a clear understanding of her car's travel capabilities within her budget.