Question

A navigator marks three locations on a satellite map as shown. If the distance from Serravalle to San Marino is 982 miles, and the distance from Chiesanuova to San Marino is 833 miles, then how far is Chiesanuova from Serravalle? In a satellite map of San Marino, Serravalle, and Chiesanuova in which navigation of three locations in the obtuse angle of  San Marino 113.23 degrees, three points are located inside the yellow mark border A. 2,300 miles B. 1,518 miles C. 1,233 miles D. 1,052 miles E. 998 miles

Answer 1

To find the distance between Chiesanuova and Serravalle, the Law of Cosines is used with given distances from each to San Marino and the included angle. The calculation results in approximately 1,518 miles, which is option B.

Step-by-step explanation:

The problem at hand involves determining the distance between Chiesanuova and Serravalle in San Marino, given the distances from each to San Marino itself and the angle at San Marino. This is a classic application of the Law of Cosines in trigonometry, which allows us to find the length of a side of a triangle when we know the lengths of the other two sides and the included angle. The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and where C is the angle opposite side c, the formula is c2 = a2 + b2 - 2ab*cos(C).

In this scenario, we are given:
a = 982 miles (distance from Serravalle to San Marino),
b = 833 miles (distance from Chiesanuova to San Marino),
C = 113.23 degrees (the angle at San Marino).

Applying the Law of Cosines:
c2 = 9822 + 8332 - 2(982)(833)cos(113.23 degrees).

Using a calculator, we can solve for c:

c = sqrt(9822 + 8332 - 2(982)(833)cos(113.23 degrees))

After calculating, we find that c is approximately 1,518 miles. Therefore, the distance between Chiesanuova and Serravalle is 1,518 miles, which corresponds to option B.