Determine how far a boat is from shore

Question

To determine how far a boat is from shore, two radar stations have been erected 1 mile apart on the shoreline. These stations then measure the angles out to the boat as shown in Figure 2. Find the (perpendicular) distance of the boat from shore. Hint First determine the distance of the boat to one of the radar stations.

Figure 1 has 370 km, 260 km, and 2.1 degrees.

Figure 2 has 1 mile, 70 degrees, and 45 degrees.

Answer 1

For Figure 2, construct a triangle

1 mile is opposite an angle = 180-(70+45) = 65 degrees

Use the Law of Sines to find the other two sides

sin65/a = sin65/1 = sin65 = sin70/b = sin45/c

b = sin70/sin65, c=sin45/sin65

b= 1.037 miles, c= 0.780 miles

b or c = distance from each radar to the boat

construct 2 right triangles with a perpendicular line

through the a,b,c triangle

1 right triangle has hypotenuse =1.036 miles, the other 0.78 miles

1.036sin45 = 0.78sin70 = 0.733 miles

perpendicular distance of the boat from the shore = 0.733 miles

For Figure 1, use the law of cosines to find the 3rd side

use law of sines to find the other 2 angles

similar to figure 2, construct right triangles to solve for the perpendicular distance