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?

Short answer

The diameter of old wire rope needed to safely lift a 9-ton load is \(D=\sqrt {4.5}\) inches.

Step-by-step solution

Rewrite the model for an old rope

Given the original model, \(4 \cdot D^{2}=S\), the safe working load \(S\) for a new rope is decreased by 50% for an old or worn rope. Therefore, the model for an old rope will be \(S'=0.5S\). Substituting the original formula into \(S'\), we get \(S'=0.5 \cdot (4 \cdot D^{2}) = 2 \cdot D^{2}\).

Solve for D using the old rope model

The problem now is to determine the diameter \(D\) of old rope needed to safely lift a 9-ton load. In other words, when \(S'=9\), what is \(D\)? Substitute \(S'=9\) into the old rope model and solve for \(D\): \(9 = 2 \cdot D^{2}\). Divide both sides by 2 to get \(D^{2}=4.5\).

Find D

To find \(D\), take the square root of both sides. Make sure to consider only the positive root since diameter is a positive quantity. Therefore, \(D=\sqrt {4.5}\).