A surveyor takes measurements as shown in the diagram. Calculate the distance across the lake. Give your answer to one decimal place.
We can see that this question involves a triangle in which we’ve been given the lengths of two sides. They’re 800 feet and 900 feet. And the measure of one angle is 70 degrees. We want to calculate the length of the third side in this triangle, which we can refer to as 𝑑 feet. The measurements we’ve been given are a specific set of information. They are two side lengths and the included angle, which tells us we can answer this question by applying the law of cosines.
This tells us that when we know two sides of a triangle, 𝑏 and 𝑐, and the included angle, capital 𝐴, we can calculate the third side of the triangle, lowercase 𝑎, using the formula 𝑎 squared equals 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 cos 𝐴. In our question, the measurements 𝑏 and 𝑐 are the side lengths of 800 and 900 feet. And the angle 𝐴 is the angle of 70 degrees. Substituting the relevant values then, and we have that 𝑑 squared is equal to 800 squared plus 900 squared minus two multiplied by 800 multiplied by 900 multiplied by cos of 70 degrees.
We could type this directly into a calculator or we could work it out in stages. 800 squared plus 900 squared is 1450000. And two multiplied by 800 multiplied by 900 is 1440000. Typing this line of working into our calculator, and we have 𝑑 squared equals 957490.9936. Now, we aren’t finished because we found 𝑑 squared, not 𝑑. So we must remember to square root both sides of this equation. Doing so gives 𝑑 equals 978.5146. It is a really common mistake though to forget this final step.
The question asks us to give our answer to one decimal place. So rounding appropriately, we have our answer to the problem. The distance across the lake found by applying the law of cosines in the given triangle is 978.5 feet.