Chapter 9.4, Problem 37E

To determine

To find the polar equation of the line that contains $\left(40,0°\right)\text{ and }\left(40,72°\right)$ .

Answer to Problem 37E

The polar equation of the line is $32.36=r\cos \left(\theta +36\right)$

Explanation of Solution

Given information:

The line contains $\left(40,0°\right)\text{ and }\left(40,72°\right)$ .

The radius of the plot is 40 feet.

Calculation:

Two coordinates must satisfy the polar equation.

Substitute the first coordinate into polar equation.

  $\begin{array}{l}p=r\cos \left(\theta -\phi \right)\\ p=40\cos \left(0°-\phi \right)\mathrm{...}\left(i\right)\end{array}$

Substitute the second coordinate into polar equation.

  $\begin{array}{l}p=r\cos \left(\theta -\phi \right)\\ p=40\cos \left(72°-\phi \right)\mathrm{...}\left(ii\right)\end{array}$

Substituting either expression for p into other equation results in the equation

  $\begin{array}{l}40\cos \left(0°-\phi \right)=42\cos \left(72°-\phi \right)\\ \phi =36\left[\text{use graphic calculator}\right]\end{array}$

Substitute the obtained value in equations i evaluate p.

  $\begin{array}{l}p=40\cos \left(70°-\phi \right)\\ p\approx 32.36\left[\phi =36\right]\end{array}$

Therefore,

The polar equation of the line is $32.36=r\cos \left(\theta +36\right)$