Question

a traveler comes upon a fork in the road. a sign on the right path reads, mercer: 24 km. a sign on the left path reads, turtle lake: 17 km. the angle between the paths is 75°, as shown. image of triangle with traveler, turtle lake, mercer, sides 17 km, 24 km, angle 75° assuming both paths are perfectly straight, how far apart are mercer and turtle lake? do not round during your calculations. round your final answer to the nearest kilometer. blank box km

Explanation:

Step1: Identify the formula

We can use the Law of Cosines to find the distance between Mercer and Turtle Lake. The Law of Cosines states that for a triangle with sides \(a\), \(b\), and \(c\) opposite angles \(A\), \(B\), and \(C\) respectively, \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\). In this case, let \(a = 24\) km, \(b=17\) km, and \(C = 75^{\circ}\), and we want to find the length of the side opposite angle \(C\) (let's call it \(c\)).

Step2: Substitute the values into the formula

Substitute \(a = 24\), \(b = 17\), and \(C=75^{\circ}\) into the Law of Cosines formula:
\[
c^{2}=24^{2}+17^{2}-2\times24\times17\times\cos(75^{\circ})
\]
First, calculate \(24^{2}=576\), \(17^{2} = 289\), and \(2\times24\times17=816\). Then, \(\cos(75^{\circ})\approx0.2588\) (we can use a calculator to find the cosine of \(75^{\circ}\)).
So,
\[
c^{2}=576 + 289-816\times0.2588
\]
\[
c^{2}=865-816\times0.2588
\]
Calculate \(816\times0.2588 = 816\times(0.2 + 0.05+0.008 + 0.0008)=816\times0.2+816\times0.05 + 816\times0.008+816\times0.0008=163.2+40.8+6.528 + 0.6528 = 211.1808\)
Then, \(c^{2}=865 - 211.1808=653.8192\)

Step3: Take the square root

To find \(c\), we take the square root of \(c^{2}\):
\[
c=\sqrt{653.8192}\approx25.57
\]
Rounding to the nearest kilometer, we get \(c\approx26\).

Answer:

26