Chapter 12.5, Problem 8P

To determine

Find the dimensions of the park.

Answer to Problem 8P

The dimensions of the park are 193.4 m and 303.6 m

Explanation of Solution

Given:

Opposite corners of a small rectangular park are joined by diagonal paths , each 360 m long.The paths intersect at a $65°$ angle.

Formula Used:

  $\sin \theta =\frac{\text{length of opposite side to }\theta }{\text{length of }hypotenuse}$

  $\cos \theta =\frac{\text{length of adjacent side to }\theta }{\text{length of }hypotenuse}$

Calculation:

Represent the given situation :

  

  

Since the diagonals intersect each other , we take the upper right triangle .

y represents the half of the length of the triangle and x represents half of the width of the rectangle.

The angle will be half of the angle between the diagonals = $\frac{65°}{2}=32.5°$

The hypotenuse is the half of each diagonal = $\frac{360}{2}=180$

Now, we find x and y and then double them to get the dimensions of the rectangle.

Use the sine and cosine formula :

  $\begin{array}{l}\sin 32.5°=\frac{\text{length of opposite side to 32}\text{.5}°}{\text{length of }hypotenuse}\\ \Rightarrow \sin 32.5°=\frac{y}{180}\\ \Rightarrow y=180\sin 32.5°\mathrm{...............}[\text{multiply each side by 180]}\\ \Rightarrow y=96.7139295\\ \Rightarrow L=2y=193.427859\\ \Rightarrow L\approx 193.4\end{array}$

  $\begin{array}{l}\cos 32.5°=\frac{\text{length of adjacent side to 32}\text{.5}°}{\text{length of }hypotenuse}\\ \Rightarrow \cos 32.5°=\frac{x}{180}\\ \Rightarrow x=180\cos 32.5°\mathrm{................}[\text{multiply each side by 180]}\\ \Rightarrow \text{W=2}x=360\cos 32.5°\\ \Rightarrow W=303.62092\approx 303.6\end{array}$

So, the dimensions of the park are 193.4 m and 303.6 m .