Question

A function \(y=f(x)\) such that if your car gets \(32 \mathrm{mi} /\) gal and gasoline costs \(\$ x /\) gallon, then \(\$ 100\) is the cost of taking a \(y\) -mile trip.

Short answer

Answer: The function representing the cost of taking a y-mile trip for a car that gets 32 miles per gallon with a gasoline price of $x per gallon is \(y = f(x) = 32 \cdot \frac{100}{x}\).

Step-by-step solution

Write down the relationships between the variables

We know that the car consumes 1 gallon for every 32 miles traveled, so the fuel consumption (in gallons) for y miles can be represented as \(y/32\). We also know that if gasoline costs \(x per gallon, then the total cost of this y-mile trip would be \)(y/32)x$.

Equate total cost to $100

Now, we need to find the function \(y = f(x)\) such that the cost of taking a y-mile trip is \(100. So, we can set the total cost equal to \)100: $$100 = \frac{y}{32} x$$

Solve for y

To solve for y and obtain the function \(y = f(x)\), we can first divide both sides of the equation by x: $$\frac{100}{x} = \frac{y}{32}$$ Now, we just need to multiply both sides by 32 to obtain the function y in terms of x: $$y=f(x) = 32 \cdot \frac{100}{x}$$

Verify the result

Let's verify that the derived function has the correct property. If we have a $100 cost for a y-mile trip, we should have: $$y = 32 \cdot \frac{100}{x}$$ Now, plugging the cost in as \(100, we should be able to find the corresponding value of y. If the total cost of the trip is \)\$100\( with the price of gasoline \)\$ x /$ gallon, then the miles traveled is y: $$y = 32 \cdot \frac{100}{x}$$ We have now found the function \(y = f(x) = 32 \cdot \frac{100}{x}\), which represents the cost of taking a y-mile trip for a car that gets 32 miles per gallon with a gasoline price of $x per gallon.

Key concepts

Fuel Consumption Rate

Understanding the fuel consumption rate is crucial for calculating the cost associated with driving a vehicle over a certain distance. The fuel consumption rate is essentially the measure of the fuel efficiency of a vehicle, indicating the distance that can be traveled per unit of fuel.

Fuel efficiency is usually stated as 'miles per gallon' (mpg) in the U.S. or as 'liters per 100 kilometers' (L/100km) in other parts of the world. In our example, the car's fuel efficiency is given as 32 miles per gallon. This means for every gallon of gas, the car can travel 32 miles.

To determine the fuel needed for a particular journey, one divides the distance to travel (y miles) by the car's fuel efficiency (32 mpg), as shown in the exercise. The resulting number shows the total gallons of gas needed for the trip. If you know the price per gallon (\(x\) dollars), multiplying it by the total gallons required gives you the total cost of fuel for the trip.

Solving for y in terms of x

The process of solving for y in terms of x involves isolating the variable y on one side of the equation, with the remaining terms on the other side represented in terms of x. This manipulation turns the equation into a function, where y becomes a dependent variable whose value is determined by the independent variable x.

In the context of our exercise, we begin with an equation that represents the total fuel cost for a y-mile trip. By employing algebraic techniques and following the provided steps, we rearranged the equation to express y solely in terms of x. The initial equation, which starts as \(100 = \frac{y}{32} x\), becomes \(y = 32 \cdot \frac{100}{x}\) after isolating y.

This operation is essential in many fields such as economics, physics, and engineering where understanding the relationship between two variables is key. By knowing how to solve for y in terms of x, one can predict outcomes and make informed decisions based on variable inputs.

Functional Relationships in Calculus

In calculus, functional relationships express how one quantity changes with respect to another. A function denoted as \(y = f(x)\) indicates that y is dependent on x. The changes in the value of x dictate the resultant changes in the value of y.

In our problem, the functional relationship is between the cost of gasoline \(x\) and the distance traveled \(y\). By establishing this relationship through a function, we are able to model real-world processes mathematically. Understanding functions allows us to analyze how one variable affects another and to calculate derivatives and integrals which reveal rates of change and accumulations, respectively.

By interpreting functional relationships using calculus, we gain the ability to solve problems that require optimization, such as minimizing costs or maximizing efficiency, which are common objectives in economics and engineering disciplines.