Chapter 3.4, Problem 49PS

Solution

Find the inverse of the model. Then find the waterline length needed to achieve a maximum speed of 7.5 knots.

  $v^{-1}={\left(\frac{l}{1.34}\right)}^2$

Waterline length needed to achieve a maximum speed (v) of 7.5 knots is $(l)=31.36$ feet

Given information:

The maximum hull speed v (in knots) of a boat with a displacement hull can be approximated by $v=1.34\sqrt{l}$ where $l$ is the length (in feet) of the boat’s waterline.

  

Concept used:

In order to find the inverse of a function $y=f\left(x\right)$, follow below steps:

Step 1: interchange x and y in the given function

Step 2: solve for y and that will give the required inverse function

Calculation:

The inverse of model $v=1.34\sqrt{l}$ is -

  $\begin{array}{c}v=1.34\sqrt{l}\\ y=v\\ y=1.34\sqrt{l}\\ swaplwithy\\ l=1.34\sqrt{y}\\ solvefory\\ l=1.34\sqrt{y}\\ \sqrt{y}=\frac{l}{1.34}\\ y={\left(\frac{l}{1.34}\right)}^2\\ v^{-1}={\left(\frac{l}{1.34}\right)}^2\end{array}$

Thus the inverse of the given function $v=1.34\sqrt{l}$ is $v^{-1}={\left(\frac{l}{1.34}\right)}^2$ .

The waterline length needed to achieve a maximum speed (v) of 7.5 knots is −

  $\begin{array}{c}v=1.34\sqrt{l}\\ 7.5=1.34\sqrt{l}\\ \sqrt{l}=\frac{7.5}{1.34}\\ l={\left(\frac{7.5}{1.34}\right)}^2\\ $ l={(5.6)}^2$l=31.36\end{array}$

So the waterline length needed to achieve a maximum speed (v) of 7.5 knots is $(l)=31.36$