Question

Elastic bands can be used for exercising to provide a range of resistance. The resistance \(R\) (in pounds) of a band can be modeled by \(R=\frac{3}{8} L-5\), where \(L\) is the total length (in inches) of the stretched band. Find the inverse function. What length of the stretched band provides 19 pounds of resistance?

Short answer

The length of the stretched band that provides 19 pounds of resistance is 64 inches.

Step-by-step solution

Calculate the inverse function

The given function is \(R=\frac{3}{8} L-5\). To find the inverse function, one must first swap \(R\) and \(L\), resulting in \(L=\frac{3}{8} R-5\). Then solving this equation for \(R\) gives the inverse function.

Solve for inverse function

Start by adding 5 to both sides of the equation to isolate the term containing \(R\), giving \(L+5=\frac{3}{8} R\). Then, multiply both sides of the equation by \(\frac{8}{3}\) which gives the inverse function, \(R = \frac{8}{3} (L+5)\).

Substitute the resistance value into the inverse function

Now that we have the inverse function, we can substitute the given resistance value \(R = 19\) pounds into the function to find the length of the stretched band. This gives us \(L = \frac{8}{3} (19+5)\).

Calculate the length of the stretched band

Evaluating the expression gives \(L = \frac{8}{3} \times 24\) or \(L =64\) inches.

Key concepts

Mathematical Modeling

Understanding how to represent real-world situations with mathematical equations is a fundamental concept in algebra known as mathematical modeling. In our exercise, the resistance offered by an elastic exercise band is modeled through the equation
\( R = \frac{3}{8} L - 5 \)
where R represents the resistance in pounds, and L is the total length of the stretched band in inches. This format allows us to input any length L and get the corresponding resistance R, which is an indispensable skill when needing to predict outcomes or understand the relationship between variables in practical scenarios.

To make the model more useful, it can be helpful to find the inverse function. This allows us to start with a desired resistance and find the needed length of a band to achieve it, effectively reversing the process. In doing so, we can address problems from both directions—either predicting resistance based on length, or determining the necessary length to produce a specific resistance. This flexibility is what makes mathematical modeling so valuable in fields ranging from engineering to finance.

Algebraic Expressions

In algebra, we frequently use algebraic expressions to represent relationships between different quantities. An algebraic expression can contain variables, constants, and operations such as addition, subtraction, multiplication, and division. In the given problem,
\( R = \frac{3}{8} L - 5 \)
is an algebraic expression where the term \( \frac{3}{8} \) is a constant coefficient, L is the variable representing length, and 5 is a constant that is subtracted from the product of \( \frac{3}{8} \) and L. Working with these expressions involves understanding how to manipulate and rearrange them to isolate specific variables—an essential skill for uncovering relationships within the problem.

To improve clarity for students in solving this exercise, attention could be paid to the process of finding the inverse of an algebraic expression. By interchanging the variables and then isolating the new 'output' variable, we demonstrate how to rewrite the expression in a form that provides us the desired set of inputs and outputs, paving the way for solving equations effectively.

Solving Equations

The process of finding the values of variables that make an equation true is known as solving equations. It is a fundamental aspect of many mathematical problems, and mastery of this skill enable students to unlock a deeper understanding of various mathematical concepts.

In our exercise, solving the original model equation for L requires algebraic manipulation, including reversing operations and maintaining balance (what you do to one side of the equation, you must do to the other side). This balancing act ensures that the equality is preserved while isolating the variable of interest, which in our scenario is to find the inverse function. Once the inverse function \( R = \frac{8}{3} (L+5) \) is established, we apply it to determine the length L that corresponds to a given resistance R.

In the context of our example, to find the length that provides 19 pounds of resistance, we plug 19 in place of R in the inverse function and simplify to get L. This step-by-step approach not only provides the solution (64 inches) but also reinforces the methodology behind solving equations, an invaluable tool in both academic and real-world problem solving.