From a mountain 1780 ft. high, the angle of depression of a point on the nearer shore of a river is 48 degrees 40 minutes and of a point directly across on the opposite side is 22 degrees 20 minutes. What is the width of the river between the two points?
If you draw a diagram, you will see that you cannot compute the width of the river without knowing how far the river is from the mountain.
If we assume that the river is at the same elevation of the base of the mountain, then
Width of the river
=distance of far shore - distance of near shore
=1780/tan(θ2)-1780/tan(θ1)
To find the width of the river between the two points, we can use the concept of trigonometry and the angle of depression measurements.
Let's break down the problem into smaller parts:
1. First, let's consider the point on the nearer shore. The angle of depression from the mountain is given as 48 degrees 40 minutes. We can refer to this as angle A.
2. Now let's consider the point on the opposite side directly across the river. The angle of depression from the mountain is given as 22 degrees 20 minutes. We can refer to this as angle B.
3. To find the width of the river, we need to find the measurements of the two angles formed by the point on the nearer shore, the top of the mountain, and the point directly across on the opposite side. Let's label these angles X and Y.
4. Since we are given the height of the mountain (1780 ft.) and we know that the points on the nearer shore and opposite side lie on the same horizontal line, we can consider triangle AYM (where Y is the point across the river and M is the top of the mountain).
5. In triangle AYM, we have a right angle at Y. The angle AYM is the complement of angle A (90 degrees - angle A). Similarly, the angle MYA is the complement of angle B (90 degrees - angle B).
6. To find the width of the river (the line segment YM), we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side.
7. In triangle AYM, we can write the tangent equation for angle AYM as: tan(AYM) = (width of the river) / (height of the mountain).
8. Similarly, for angle MYA, we can write the tangent equation as: tan(MYA) = (width of the river) / (height of the mountain).
9. Rearranging these equations, we have: (width of the river) = (height of the mountain) * tan(AYM) and (width of the river) = (height of the mountain) * tan(MYA).
10. Now we need to calculate the values of tan(AYM) and tan(MYA).
Using the given angle A (48 degrees 40 minutes), we can convert it to decimal form: 48 + 40/60 = 48.67 degrees.
Similarly, using the given angle B (22 degrees 20 minutes), we can convert it to decimal form: 22 + 20/60 = 22.33 degrees.
11. Once we have the decimal values for angles AYM and MYA, we can use a scientific calculator to find their tangent values.
12. Using the tangent values and the height of the mountain (1780 ft.), we can substitute them into the equations from step 9 and calculate the width of the river.
Following these steps, you should be able to calculate the width of the river between the two given points.