Problem 61 - 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: \(4 \cdot D^{2}=S\) When determining the safe working load \(S\) of a rope that is old or worn, decrease \(S\) by \(50 \% .\) Write a model for \(S\) when using an old wire rope. What diameter of old wire rope do you need to safely lift a 9 -ton load?

Step-by-Step Solution

Answer

The diameter of an old wire rope that can safely lift a 9-ton load is given by \(D\approx\sqrt{4.5}\) inches.

Step 1: Formulate New Model For Old Wire Rope

The initial model for the safe working load of a new wire rope is \(4 \cdot D^{2}=S\). For an old wire rope, the safe working load decreases by 50%. Hence, the model for the safe working load \(S_{old}\) of an old wire rope is \(S_{old} = 0.5 \cdot 4 \cdot D^{2}=2 \cdot D^{2}\).

Step 2: Substitute Load into Equation and Solve for D

It's asked to find the diameter of an old wire rope that can safely lift a 9-ton load. We plug \(S_{old}=9\) into the equation to solve for \(D\). Hence, the equation is \(9 = 2 \cdot D^{2}\). If we divide both sides by 2, we get \(D^{2}=\frac{9}{2}=4.5\). Calculate the square root on both sides to solve for \(D\).

Step 3: Calculating Diameter Of Wire Rope.

To find out the diameter, take the square root on both sides of the equation. When you do this, \(D=\sqrt{4.5}\). Calculate the square root of 4.5 to get the diameter.

Key Concepts

Algebraic Modeling

Algebraic modeling is a mathematical approach that uses algebra to represent real-world situations. It transforms a given scenario into an equation or a system of equations, enabling one to solve for unknown variables. In the context of wire rope safe working load, algebraic modeling is used to construct a functional relationship between the diameter of the wire rope and its safe working load.

The initial model provided is the equation \(4 \cdot D^2 = S\), indicating that the safe working load (\(S\)) is directly proportional to the square of the diameter (\(D\)) with a constant factor of four. When the wire rope is deemed old or worn, the scenario changes, and this necessitates a new model. The safe working load must be reduced by 50%, and thus the model is adjusted to \(2 \cdot D^2 = S_{old}\), where \(S_{old}\) represents the safe working load of the aged rope. This revision algebraically captures the deterioration in rope strength due to aging or wear.

Square Root

The square root is a fundamental operation in mathematics that answers the question, 'What number, when multiplied by itself, will produce the given value?' It is represented by the radical symbol \(\sqrt{}\) and is crucial in solving quadratic equations where one must find the value of a variable squared.

Typically, in exercises involving wire ropes and load capacities, once the equation is formulated, finding the diameter may involve solving for the square of the diameter (\(D^2\)). To extract \(D\) from its squared form, the square root operation is employed. For instance, when given \(D^2 = 4.5\), calculating the square root on both sides (\