Chapter 4.2, Problem 66E

Solution

To identify: The Allen’s errors and correct them. If Allen’s Garden is a quarter-circle with a radius of $10ft$ . He wishes to plant his garden in four parallel strips, as shown in the diagram below, so that the four arcs along the circular edge of the garden are all of equal length. After measuring four equal arcs, he carefully measures the widths of the four strips and records his data in the table shown at the right below.

Alicia sees Allen’s data and realizes that he could have saved himself some work by figuring out the strip widths via trigonometry. By checking his data with a calculator she is able correct two measurement errors he has made.

  

  $\begin{array}{cc}Strip& Width\\ A& 3.83ft\\ B& 3.34ft\\ C& 2.07ft\\ D& 0.76ft\end{array}$

The both measurement $B\text{ and }C$ were off by a tenth of a foot.

Given information:

The given table is:

  $\begin{array}{cc}Strip& Width\\ A& 3.83ft\\ B& 3.34ft\\ C& 2.07ft\\ D& 0.76ft\end{array}$

Explanation:

Consider the given information:

Using the value of $\theta$ as derived to the left we can calculate the ratio between the radius and the adjacent side of the formed right triangle by utilizing the cosine function.

  $\begin{array}{l}\cos 3\theta =\frac{A}{r}\\ \cos 2\theta =\frac{A+B}{r}\\ \cos \theta =\frac{A+B+C}{r}\\ r=A+B+C+D\end{array}$

Definition of the cosine function for arbitrary radius $r$ Fourth equation is a trivial notion.

  $\begin{array}{c}A=\cos 3\theta r\\ A+B=\cos 2\theta r\\ A+B+C=\cos \theta r\\ A+B+C+D=r\end{array}$

Rearranging the above equations.

Substitute the first equation into the second the third and the third into the forth.

  $\begin{array}{c}A=\cos 3\theta r\\ \left(\cos 3\theta r\right)+B=\cos 2\theta r\\ \left(\cos 2\theta r\right)+C=\cos \theta r\\ \left(\cos \theta r\right)+D=r\end{array}$

Rearrange each for $A,B,C,D$

  $\begin{array}{c}\cos 3\theta r=A\\ \left(\cos 2\theta -\cos \theta \right)r=B\\ \left(\cos \theta -\cos 2\theta \right)r=C\\ \left(1-\cos \theta \right)r=D\end{array}$

Evaluate for given/derived values:

  $r=10\text{and}\theta =22.5$

  $\begin{array}{l}A=10\cos 67.5°\approx 3.829ft\\ B=10\left(\cos 45°-\cos 67.5°\right)\approx 3.244ft\\ C=10\left(\cos 22.5°-\cos 45°\right)\approx 2.168ft\\ D=10\left(1-\cos 22.5°\right)\approx 0.761ft\end{array}$

Both measurements $B\text{ and }C$ were off by a tenth of a foot.