Question

You need 50 pounds of two commodities costing \(\$ 1.25\) and \(\$ 1.60\) per pound. (a) Verify that the total cost is \(y=1.25 x+1.60(50-x)\), where \(x\) is the number of pounds of the less expensive commodity. (b) Find the inverse function of the cost function. What does each variable represent in the inverse function? (c) Use the context of the problem to determine the domain of the inverse function. (d) Determine the number of pounds of the less expensive commodity purchased if the total cost is \(\$ 73 .\)

Short answer

For the total cost \$73, you need to buy approximately 27 pounds of the less expensive commodity and the remaining about 23 pounds of the more expensive one.

Step-by-step solution

Verify the Total Cost Equation

Starting with the given information, you need a total of 50 pounds comprising two commodities. Let 'x' be the amount of the less expensive commodity (costing \$1.25/pound). Then the amount of the more expensive commodity (costing \$1.60/pound) is (50-x). Thus, the total cost equation can be formulated as \(y = 1.25x + 1.60(50-x)\)

Calculate the Inverse Function

Rewrite the total cost equation in terms of 'x'. The equation becomes \(x = \frac{y - 1.60(50)}{1.25-1.60}\). This is the inverse of the total cost function. Here, 'y' represents the total cost and 'x' is the number of pounds of the less expensive commodity.

Derive the Domain of the Inverse Function

Since 'x' represents the quantity of a commodity, it cannot be less than 0. Also, because the total quantity cannot exceed 50 pounds, 'x' cannot be greater than 50. Thus, the domain of the inverse function is \(0 \leq x \leq 50\)

Determine Quantity with Specific Total Cost

If the total cost is \$73, then substitute 'y' with 73 in the inverse function to solve for 'x', that is the amount of the less expensive commodity you should purchase. Solving gives \(x = \frac{73 - 1.60(50)}{1.25-1.60}\)

Key concepts

Understanding Inverse Functions

The concept of inverse functions is a fundamental one in calculus and algebra that arises when we have two variables related in such a way that one can be expressed as a function of the other. For instance, if we know the cost function, which tells us the total cost based on the quantity purchased, the inverse function will tell us the quantity that corresponds to a certain total cost.

In the context of our textbook problem, the given cost function is represented by the equation \( y=1.25x+1.60(50-x) \), where \( y \) is the total cost and \( x \) is the amount of the less expensive commodity purchased. To find the inverse, we solve this equation for \( x \) in terms of \( y \) which gives us the ability to determine the quantity of commodity purchased (\( x \) in this case) for any given total cost \( y \).

This inversion is more than just an algebraic trick. It is incredibly useful in real-world situations. When dealing with prices and budgets, you often know how much money you have (the total cost, \( y \)) and need to figure out how much of a product or service you can afford (the quantity, \( x \)). The inverse function facilitates this, turning a question of economics into a straightforward mathematical solution.

Determining the Domain of a Function

Understanding the domain of a function is essential as it defines the set of possible inputs for which the function is defined. Simply put, the domain consists of all the values that we can put into the function without causing an error in calculation, such as division by zero or taking the square root of a negative number.

In our example of commodities, the domain of the inverse function is particularly important. It is constrained by physical limitations: the number of pounds of the commodity,\( x \), cannot be negative and you cannot have more than 50 pounds because that's the total weight you need. Therefore, the domain of our inverse function is \( 0 \leq x \leq 50 \). It's critical to specify this domain when solving practical problems because it adheres to the real-world scenario we are modeling. When considering the inverse function, it makes no sense to talk about negative pounds of commodities or amounts that exceed the total required.

Especially when we look for solutions within contexts like budgeting or buying supplies, defining these boundaries prevents impossible or absurd solutions, such as buying a negative amount of something or more than you intended.

Working with Linear Equations

Linear equations are the simplest type of algebraic function and form the backbone of many algebraic concepts, including the calculus problem we're examining. A linear equation is one in which two variables are related in a straight line when graphed, and it can be written in the form \( y=mx+b \), where \( m \) is the slope and \( b \) is the y-intercept.

Our problem involves a linear equation that represents the total cost as a function of the weight of the commodities purchased. The coefficients (1.25 and 1.60) represent the cost per pound of the two different commodities, and the structure of the equation \( y = 1.25x + 1.60(50-x) \) confirms its linear nature. Solving linear equations is usually straightforward and involves isolating the variable of interest on one side of the equation.

To connect this to the concept of inverse functions, for obtaining the inverse function, we essentially 'solve' the equation for \( x \), treating \( y \) as the known value. By manipulating the equation in this way, we discover a new linear relationship, which now shows the quantity of the commodity as a function of the total cost. Remembering the properties of linear equations can shine a light on how to tackle them and use them to model real-life situations effectively.