Chapter 9.7, Problem 9CYU

To determine

To find: The position of the cellar.

Answer to Problem 9CYU

The exact location is $\left(40,30\right)$ .

Explanation of Solution

Given:

The one unit represents the one mile, the location of the caller is determined to be 50 miles from the tower of the origin.

The person is 40 miles from a tower at (0,30) and 13 miles from the tower $\left(35,18\right)$ .

Calculation:

The distance of the caller from the tower at the origin by the distance formula is,

  $\begin{array}{l}\sqrt{{\left(h-0\right)}^2+{\left(k-0\right)}^2}=50\\ h^2+k^2=2500\end{array}$

Consider the distance of the caller from the tower of the point (0, 30) by the distance formula as,

  $\begin{array}{l}\sqrt{{\left(h-0\right)}^2+{\left(k-30\right)}^2}=40\\ {\left(h-0\right)}^2+{\left(k-30\right)}^2=1600\\ h^2+{\left(k-30\right)}^2=1600\end{array}$

Consider the distance of the caller from the tower of the point (35, 18) by the distance formula as,

  $\begin{array}{l}\sqrt{{\left(h-35\right)}^2+{\left(k-18\right)}^2}=13\\ {\left(h-35\right)}^2+{\left(k-18\right)}^2=169\end{array}$

Subtract the second equation from the first as,

  $\begin{array}{l}{h}^2+k^2-h^2-{\left(k-30\right)}^2=2500-1600\\ k^2-{\left(k-30\right)}^2=900\\ 60k-900=900\\ k=30\end{array}$

Substitute $k=30$ in the equation $h^2+k^2=2500$ as,

  $\begin{array}{l}{h}^2+{\left(30\right)}^2=2500\\ h=\pm 40\end{array}$

Thus, the solution for the equations $h^2+k^2=2500$ and $h^2+{\left(k-30\right)}^2=1600$ are $\left(40,30\right),\left(-40,30\right)$ .

Substitute the points in the equation ${\left(h-35\right)}^2+{\left(k-18\right)}^2=169$ as,

  $\begin{array}{l}{\left(40-35\right)}^2+{\left(30-18\right)}^2=169\\ 25+144=169\\ 169=169\end{array}$

Also,

  $\begin{array}{l}{\left(-40-35\right)}^2+{\left(30-18\right)}^2=169\\ 5625+144=169\\ 5769\ne 169\end{array}$

Thus, the exact location is $\left(40,30\right)$ .