Chapter 4.6, Problem 63E

To determine

Sketch the graph of the function over the interval $0<x<\pi .$

Answer to Problem 63E

  $d=7\cot x$

Explanation of Solution

Given information: Altitude = 7 miles, d = ground distance from antenna to point directly under the plane and x be the angle of elevation to the plane from antenna.

Formula Used: $\cot x=\frac{adjacent\_side}{opposite\_side}$

Calculation:

The altitude of the plane is 7 miles.

The ground distance from the antenna to the point directly under the plane is d.

The angle of elevation to the plane from the antenna is x.

The objective is to express d as a function of x and sketch its graph in interval $0<x<\pi .$ .

Represent the above information in the following figure:

  

The triangle in the figure is a right angled triangle.

Use the following definition of cotangent function.

  $\cot x=\frac{adjacent\_side}{opposite\_side}$

Substitute the lengths of the sides from the figure and write the equation as follows:

  $\cot x=\frac{adjacent\_side}{opposite\_side}$

  $\cot x=\frac{d}{7}$

Solve for d as follows:

  $d=7\cot x$

Thus, the required expression for d as a function of x is $d=7\cot x$ .

Graph of $7\cot x$