Chapter 8.7, Problem 41PS

Solution

i.

To find: To find the equation that represents the circle and Oak Lane.

The equation that represents the circle and Oak Lane are $x^2+y^2\le 1\text{ and }y=-\frac{1}{7}x+\frac{5}{7}$ .

Given information: The radius of the circle is 1 mile and the origin are at $(0,0)$ .

Calculation:

The radius of the circle is 1 mile.

The parking pass is given by,

  $x^2+y^2\le 1$

The coordinates of Oak Lane are $(-2,1)\text{ and }(5,0)$

Using the formula,

  $y=kx+l$

Substitute the coordinate value $\text{(5, 0)}$ in the above formula,

  $\begin{array}{c}y=kx+l\\ 0=5k+l\\ l=-5k\end{array}$

  $$

Substitute the coordinate value $\text{(-2, 1)}$ in the above formula,

  $\begin{array}{c}y=kx+l\\ 1=-2k-5k\\ 1=-7k\\ k=-\frac{1}{7}\end{array}$

Put $k=-\frac{1}{7}$ in $l=-5k$

  $l=\frac{5}{7}$

Hence,

  $y=-\frac{1}{7}x+\frac{5}{7}$

Conclusion:

The equation for the circle and Oak Lane is $x^2+y^2\le 1\text{ and }y=-\frac{1}{7}x+\frac{5}{7}$ .

ii.

To find: To solve the circle and Oak Lane equations.

The plots of the line are from $\left(-\frac{3}{5},\frac{4}{5}\right)\text{ to }\left(\frac{4}{3},\frac{3}{5}\right)$ .

Given information: The equation that represents the circle and Oak Lane are $x^2+y^2\le 1\text{ and }y=-\frac{1}{7}x+\frac{5}{7}$ .

Calculation:

The equation that represents the circle and Oak Lane are $x^2+y^2\le 1\text{ and }y=-\frac{1}{7}x+\frac{5}{7}$ .

Now substituting the values of $x$ to get the values of $y$ ,

  $\begin{array}{c}{y}_1=-\frac{1}{7}{x}_{{}_1}+\frac{5}{7}\\ y_1=\frac{4}{5}\end{array}$

  $\begin{array}{l}{y}_{{}_2}=-\frac{1}{7}{x}_{{}_2}+\frac{5}{7}\\ y_{{}_2}=\frac{3}{5}\end{array}$

Conclusion:

The plots of the line are from $\left(-\frac{3}{5},\frac{4}{5}\right)\text{ to }\left(\frac{4}{3},\frac{3}{5}\right)$ .

iii.

To find: To find the length of the Oak Lane for the students who are not eligible for the parking pass.

The length of the Oak Lane for the students who are not eligible for the parking pass is $1.41$ miles.

Given information: Non eligible students for the parking pass intersect with the circle at a radius of 1.

Calculation:

The distance between the two points is given by,

  $\begin{array}{c}d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(0.8+0.6\right)}^2+{\left(0.6-0.8\right)}^2}\\ =\sqrt{{\left(1.4\right)}^2+{\left(-0.2\right)}^2}\\ =\sqrt{2}\\ \approx 1.41\end{array}$

Conclusion:

The length of the Oak Lane for the students who are not eligible for the parking pass is $1.41$ miles.