To indirectly measure the distance across a river, Chee stands on one side of the river and uses sight-lines to a landmark on the opposite bank. Chee draws the diagram below to show the lengths and angles that she measured. Find P, RPR, the distance across the river. Round your answer to the nearest foot.
To find the distance across the river, we can use the Law of Sines.
sin(P) / PR = sin(29°) / 200
sin(P) / PR = 0.4848
sin(P) = 0.4848 * PR
sin(R) / PR = sin(66°) / 200
sin(R) / PR = 0.9135
sin(R) = 0.9135 * PR
Since P + R = 90°
sin(P) = cos(R)
0.4848 * PR = sqrt(1 - 0.9135^2) * PR
0.4848 = 0.4063
P ≈ 33.5°
Now we can use the Law of Sines again to find PR:
sin(33.5°) / PR = sin(66°) / 200
0.5592 / PR = 0.9135
PR = 0.5592 / 0.9135 * 200
PR ≈ 122.3 ft
Therefore, the distance across the river is approximately 122 feet.