Problem 60 - Algebra 1: Concepts and Skills

Question

The safe working load \(S\) (in tons) for a wire rope is a function of \(D,\) the diameter of the rope (in inches). Safe working load model for wire rope: \(4 \cdot D^{2}=S\) What diameter of wire rope do you need to lift a 9 -ton load and have a safe working load?

Step-by-Step Solution

Answer

The diameter of wire rope needed to safely lift a 9-ton load is 1.5 inches.

Step 1: Understand the problem

We are given the function \(4 \cdot D^{2} = S\), where \(S\) is given as 9. We need to rearrange this equation to solve for \(D\) and substitute \(S\) with 9 to find the solution.

Step 2: Rearrange the equation

To find the diameter, rearrange \(4 \cdot D^{2} = S\) to \(D = \sqrt{S / 4}\). Thus our new function to find the diameter is \(D = \sqrt{S / 4}\)

Step 3: Substitute and compute

Substitute \(S\) with 9 (the load we are lifting) into the new function: \(D = \sqrt{9 / 4}\) = 1.5. We find that the diameter required is 1.5 inches to be able to safely lift a 9-ton load.

Key Concepts

Functions

In mathematics, a function is a relationship between two variables where each input (often represented as \(x\)) has exactly one output (often represented as \(y\)). Functions can be linear, quadratic, or even more complex. A function is typically written in the form \(y = f(x)\), which means that \(y\) is a function of \(x\). Functions are crucial because they help us model real-world situations.

• This exercise involves the function \(4 \cdot D^2 = S\), where \(S\) is the safe working load and \(D\) is the diameter of the rope.
• The function relates the diameter of a wire rope to its safe working load capacity in tons.

By understanding this relationship, we can solve for any variable given the others, providing essential applications in fields like engineering and physics.

Rearranging Equations

Rearranging equations is a foundational skill in algebra that allows you to solve for a particular variable. To rearrange an equation, you manipulate it by performing operations that maintain equality. It's like organizing your room; everything is adjusted until things fit properly.

• Consider the original equation \(4 \cdot D^2 = S\). Here, we want to solve for \(D\).
• We divide both sides by 4 to isolate \(D^2\), giving us \(D^2 = \frac{S}{4}\).
• Taking the square root of both sides, we rearrange to \(D = \sqrt{\frac{S}{4}}\).

Rearranging allows you to express one variable in terms of another, making it easier to substitute values and solve practical problems. It's a critical skill that simplifies complex calculations.

Square Roots

The square root is a mathematical operation that helps find a number which, when multiplied by itself, gives the original number. The symbol for the square root is \(\sqrt{}\). Square roots are often used in algebra to simplify expressions and solve equations.

• For example, the square root of 4 is 2 because \(2 \times 2 = 4\).
• In our rearranged function \(D = \sqrt{\frac{S}{4}}\), we must find the square root of \(\frac{9}{4}\) to determine \(D\).
• Calculating this, we find \(\sqrt{\frac{9}{4}} = \frac{3}{2} = 1.5\), meaning the rope's diameter must be 1.5 inches to safely lift a 9-ton load.

Understanding square roots is vital in algebra because they allow us to handle equations involving quadratic terms and return to basic linear terms.