Question

applying the law of cosines to surveying
dan is standing 25 feet from a lamppost that is to his left and 30 feet from a lamppost that is to his right. the distance between the two lampposts is 20 feet. what is the measure of the angle formed from the two lines (each connecting to a lamppost) to the nearest degree?
\square degrees

Explanation:

Step1: Identify the Law of Cosines formula

The Law of Cosines for a triangle with sides \(a\), \(b\), \(c\) and the angle \(C\) opposite side \(c\) is \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\). In this problem, we want to find the angle \(C\) (the angle at Dan) where \(a = 25\), \(b=30\), and \(c = 20\). So we can rearrange the formula to solve for \(\cos(C)\): \(\cos(C)=\frac{a^{2}+b^{2}-c^{2}}{2ab}\)

Step2: Substitute the values into the formula

Substitute \(a = 25\), \(b = 30\), and \(c=20\) into the formula:
\(\cos(C)=\frac{25^{2}+30^{2}-20^{2}}{2\times25\times30}\)
First, calculate the numerator: \(25^{2}=625\), \(30^{2} = 900\), \(20^{2}=400\). So \(625 + 900-400=1125\)
Then, calculate the denominator: \(2\times25\times30 = 1500\)
So \(\cos(C)=\frac{1125}{1500}=0.75\)

Step3: Find the angle \(C\)

To find the angle \(C\), we take the inverse cosine (arccos) of \(0.75\): \(C=\arccos(0.75)\)
Using a calculator, \(\arccos(0.75)\approx41.41^{\circ}\) (rounded to the nearest degree)

Answer:

\(41\) (or approximately \(41\) degrees, if we round \(41.41\) to the nearest degree)