To indirectly measure the distance across a river, Makayla stands on one side of the river and uses sight-lines to a landmark on the opposite bank. Makayla 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. Since we have a right triangle, we know that the sine of angle R is equal to opposite side PR divided by the hypotenuse P.
sin(53) = PR / P
We can rearrange the equation to solve for PR:
PR = P * sin(53)
Now, we need to find the hypotenuse P using the law of sines. The sine of angle P is equal to opposite side PR divided by the hypotenuse P.
sin(74) = PR / P
We can rearrange the equation to solve for P:
P = PR / sin(74)
Plugging in the value we found earlier for PR, we get:
P = (P * sin(53)) / sin(74)
To solve for P, we can multiply both sides by sin(74) to isolate P:
P * sin(74) = P * sin(53)
sin(74) * P = sin(53) * P
sin(74) = sin(53)
P = sin(53) / sin(74)
P = 0.798 / 0.965
P = 0.826
Lastly, we can find the distance across the river, PR:
PR = P * sin(53)
PR = 0.826 * sin(53)
PR ≈ 0.826 * 0.798
PR ≈ 0.6598
Therefore, the distance across the river is approximately 0.660 feet.