Question

Approximating the Area of a Lake To approximate the area of a lake, Cindy walks around the perimeter of the lake, taking the measurements shown in the illustration. Using this technique, what is the approximate area of the lake?

[Hint: Use the Law of Cosines on the three triangles shown and then find the sum of their areas.]

Short answer

The approximate area of the lake is$8798.77\mathrm{square}\mathrm{units}.$

Step-by-step solution

Step 1. Given Information

The given figure is

Step 2. Determine side CA

In $∆ABC$, use the law of Cosine

$$\begin{gathered} A{C}^2=\frac{{125}^2+{100}^2-2\left(125\right)\left(100\right)}{\cos {50}^{\circ }} \\ AC=97.7513 \end{gathered}$$

Step 3. Determine side AD

In $∆ADE$

$$\begin{gathered} A{D}^2=\frac{{70}^2+{50}^2-2\left(70\right)\left(50\right)}{\cos {100}^{\circ }}=8615.5372 \\ AD=92.8199 \end{gathered}$$

Step 4. Determine area of each triangle

For $∆ABC$

$A_1=\frac{1}{2}\left(125\right)\left(100\right)\sin {50}^{\circ }=4787.7778$

For $∆ADE$

$A_2=\frac{1}{2}\left(70\right)\left(50\right)\sin {100}^{\circ }=1723.4136$

For $∆ACD$Use Herons Formula

$A_3=2287.5803$

Step 5. Finding area of lake

$A=4787.7778+1723.4136+2287.5803=8789.7717$Add all the area of the triangle to get the area of the lake