Chapter 9.4, Problem 35E

a.

To determine

To draw a diagram of the plot of land. Include the locations of two stakes.

Explanation of Solution

Given information:

The stakes are placed at $\left(125,130°\right)\text{ and }\left(300,70°\right)$

The radius of the plot is 500 feet.

Graph:

The diagram of the plot of land.

  

Interpretation:

Above graph shows the plot of land of radius 500 feet with two stakes placed at $\left(125,130°\right)\text{ and }\left(300,70°\right)$ .

b.

To determine

To find the polar equation of the line determined by the stakes.

Answer to Problem 35E

The polar equation of the line determined by stakes is $100=r\cos \left(\theta +0.6\right)$

Explanation of Solution

Given information:

The stakes are placed at $\left(125,130°\right)\text{ and }\left(300,70°\right)$

The radius of the plot is 500 feet.

Calculation:

Two coordinates must satisfy the polar equation.

Substitute the first coordinate into polar equation.

  $\begin{array}{l}p=r\cos \left(\theta -\phi \right)\\ p=125\cos \left(130°-\phi \right)\mathrm{...}\left(i\right)\end{array}$

Substitute the second coordinate into polar equation.

  $\begin{array}{l}p=r\cos \left(\theta -\phi \right)\\ p=300\cos \left(70°-\phi \right)\mathrm{...}\left(ii\right)\end{array}$

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

  $\begin{array}{l}125\cos \left(130°-\phi \right)=300\cos \left(70°-\phi \right)\\ \phi =-0.6\left[\text{use graphic calculator}\right]\end{array}$

Substitute the obtained value in equations i and ii and evaluate p.

  $\begin{array}{l}p=300\cos \left(70°-\phi \right)\\ p\approx 100\left[\phi =-0.6\right]\end{array}$

And

  $\begin{array}{l}p=125\cos \left(130°-\phi \right)\\ p\approx -81.34\left[\phi =-0.6\right]\end{array}$

Since, p is the length of normal it must always be positive.

Therefore,

The polar equation of the line determined by stakes is $100=r\cos \left(\theta +0.6\right)$