Question

A surveyor measures the distance across a river that flows straight north by the following method. Starting directly across from a tree on the opposite bank, the surveyor walks 100 m along the river to establish a baseline. She then sights across to the tree and reads that the angle from the baseline to the tree is 35°. How wide is the river?

a) 57.5 m
b) 68.2 m
c) 82.0 m
d) 96.2 m

Answer 1

By using the tangent of the angle which is 35° and the baseline of 100 m, we calculate the width of the river to be approximately 70.0 m. Since this is not an option given in the question, we select the closest given measurement, which is 68.2 m (option b).

Step-by-step explanation:

To determine the width of the river, we can apply trigonometric principles. The surveyor's baseline of 100 m and her angle of sight at 35° to the tree form a right-angled triangle. We are to find the side opposite the 35° angle, which is the width of the river. This side can be found using the tangent function (tan) in trigonometry because tan(θ) = opposite/adjacent where θ is the angle in question.

Using the equation:

tan(35°) = width of the river / 100 m

Thus, width of the river is:

100 m * tan(35°)

After calculating this expression, the width of the river is found to be approximately 70.0 m, which is not one of the options given in the question. Therefore, we need to reconsider the given options: a) 57.5 m, b) 68.2 m, c) 82.0 m, and d) 96.2 m. With a tangent of 35° being about 0.7, multiplying by 100 gives 70 m. Hence option b, 68.2 m, is the closest to our calculation and likely to be the measurement error.