Chapter 4.4, Problem 73AYU

(a)

To determine

To calculate:

The amount of stretch

Answer to Problem 73AYU

The amount of stretch is $39ft$ .

Explanation of Solution

Given information:

  $\begin{array}{l}W=150pound\\ L=42ft\\ K=16pounds/ft\end{array}$

Formula Used:

The stiffness of the cord is calculated as:

  $K=\frac{2W\left(S+L\right)}{S^2}$

Where,

Weight of the jumper: $W$

Stretch: $S$

Free cord length: $L$

Calculation:

The stiffness of the cord is calculated as:

  $K=\frac{2W\left(S+L\right)}{S^2}$

Put the given data to find amount of stretch:

  $\begin{array}{l}K=\frac{2W\left(S+L\right)}{S^2}\\ 16=\frac{2\left(150\right)\left(S+42\right)}{S^2}\\ 16S^2-300S-12600=0\\ 4S^2-75S-3150=0\end{array}$

The roots of the equation find out by using the quadratic formula: $a=4,b=-75,c=-3150$

  $\begin{array}{l}S=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ S=\frac{-\left(-75\right)\pm \sqrt{{\left(-75\right)}^2-4\left(4\right)\left(-3150\right)}}{2\left(4\right)}\\ S=\frac{75\pm 236.7}{8}\\ S_1=\frac{75+236.7}{8}OR\text{}{S}_2=\frac{75-236.7}{8}\\ S_1=38.962\approx 39S_2=-20.212\end{array}$

The amount of stretch is never negative so $S=39ft$ .

(b)

To determine

To calculate:

The minimum required height for the ledge in the part (a)

Answer to Problem 73AYU

The minimum required height is $84ft$ .

Explanation of Solution

Given information:

  $\begin{array}{l}W=150pound\\ L=42ft\\ K=16pounds/ft\\ S=39ft\end{array}$

Know that:

The minimum distance to the ground together is $3ft$ .

The ledge height is calculated by adding the amount of the stretch, rope length and minimum distance to the ground together.

The ledge height is safe minimum distance which allows the jumper to jump safely.

  $l=39+42+3=84$

Hence, the ledge height must be $84ft$ .