Chapter 8.1, Problem 81AYU

Explain how you would measure the width of the Grand Canyon from a point on its ridge.

To determine

To explain: How to measure the width of the Grand Canyon from a point on its ridge.

Answer to Problem 81AYU

It has been determined that the width of the Grand Canyon is the product of the height of the point on the ridge and the cotangent of the angle of depression from that point.

Explanation of Solution

Given:

A point on a ridge of the Grand Canyon

Concept used:

In a right-angled triangle, the ratio of the perpendicular and the base is equal to the tangent of the angle.

Calculation:

The given situation can be modeled by the extremely simplified figure below:

  

Let B be the given point on a ridge of the Grand Canyon, C be the farthest point from B on the opposite side at a perpendicular direction at the bottom of the Canyon and A be the intersection of the normal drawn from B horizontally and from C vertically.

Then, by our construction, $\angle CAB=90°$ and $\Delta ABC$ is a right-angled triangle.

Let $\theta$ be the angle of depression formed when one looks towards C from B. This must be known since the point B is given.

Again, since the point B is given, the height of B must be known. From the given figure, this height is clearly equal to $AC=h$ .

The length $AB=x$ must be the width of the Canyon, which we are required to find.

Now, in $\Delta ABC$ , the angle $\theta$ is known, $AC=h$ is known and is the perpendicular with respect to $\theta$ , $AB=x$ is unknown and is the base with respect to $\theta$ .

Then, by Trigonometric formula,

  $\tan \theta =\frac{h}{x}$

Simplifying,

  $x=\frac{h}{\tan \theta }$

Equivalently,

  $x=h\cot \theta$

This implies that the width of the Grand Canyon in the product of the height of the point on the ridge and the cotangent of the angle of depression from that point.

Conclusion:

It has been determined that the width of the Grand Canyon in the product of the height of the point on the ridge and the cotangent of the angle of depression from that point.