Final answer:
The problem requires solving for the width of a river using trigonometry, specifically, the tangent function applied to angles of depression from a height of 1250 feet, resulting in two right triangles to solve for the overall river width.
Step-by-step explanation:
The student's question involves using trigonometric principles to calculate the width of a river given the angles of depression from a sightseer's vantage point and the depth of the canyon. To solve this problem, we can use the concept of right-angled triangles and trigonometric ratios. The angles given are the angles of depression, which correspond to the angles of elevation from the base to the sightseer's eyes if we imagine horizontal lines extending from the sightseer to the canyon's walls.
Let's denote the width of the river as w. We will divide the problem into solving two right triangles. The canyon depth of 1250 feet serves as the opposite side for both triangles, while the widths to the near and far banks of the river serve as the adjacent sides. The trigonometric ratio relevant here is the tangent, which relates angles to the opposite and adjacent sides of a right triangle.
Using the tangent function:
- tan(61°) = opposite/near bank width
- tan(63°) = opposite/far bank width
Let x be the width from the sightseer's position perpendicular to the canyon wall on the near side, and (w - x) be the width from that same position to the far bank. Thus, we have:
- tan(61°) = 1250/x
- tan(63°) = 1250/(w - x)
Solving these equations for x and w - x allows us to calculate the river's width by finding the sum of these two distances (w = x + (w - x)).