In the town of Trigonometry, the grocery store is located 14 miles from the book store. The bookstore is located 10 miles from the school and the school is located 20 miles from the grocery store. Find the angle in degrees that the book store makes with the grocery store and school.
To find the angles, we can use the Law of Cosines. Let's label the distances d1 (14 miles), d2 (10 miles), and d3 (20 miles). Let x be the angle between the grocery store and the book store, and y be the angle between the school and the book store.
From the Law of Cosines, we have:
d1^2 = d2^2 + d3^2 - 2*d2*d3*cos(x)
Plugging in the values, we get:
14^2 = 10^2 + 20^2 - 2*10*20*cos(x)
196 = 100 + 400 - 400*cos(x)
96 = -400*cos(x)
cos(x) = -96/400
cos(x) = -0.24
x = cos^(-1)(-0.24) ≈ 103.4 degrees
So, the angle between the grocery store and the book store is approximately 103.4 degrees.
Similarly, we can find the angle between the book store and the school using the same method:
d2^2 = d1^2 + d3^2 - 2*d1*d3*cos(y)
Plugging in the values, we get:
10^2 = 14^2 + 20^2 - 2*14*20*cos(y)
100 = 196 + 400 - 560*cos(y)
100 = 596 - 560*cos(y)
-496 = -560*cos(y)
cos(y) = -496/-560
cos(y) = 0.886
y = cos^(-1)(0.886) ≈ 28.7 degrees
So, the angle between the book store and the school is approximately 28.7 degrees.