Chapter 3.1, Problem 41E

To determine

To find: coordinates at which Juana live relative to the midpoint between the two towers.

Answer to Problem 41E

  $\left(-4\sqrt{2},6\right),\left(4\sqrt{2},6\right)$

Explanation of Solution

Given information:

The equation representing the path interference of waves emitted from two towers is as follows:

  $\frac{y^2}{12}-\frac{x^2}{16}=1$

Origin is the mid-point of the line segment between the two towers.

Positive y-axis represents north. Juana lives on 6 miles north of the x-axis.

Calculation:

6 miles north of the x-axis means the y-value. Put $y=6$ in the given equation and simplify as follows:

  $\begin{array}{c}\frac{y^2}{12}-\frac{x^2}{16}=1\\ \frac{6^2}{12}-\frac{x^2}{16}=1\\ \frac{36}{12}-1=\frac{x^2}{16}\\ 3-1=\frac{x^2}{16}\\ 2=\frac{x^2}{16}\\ x^2=2\left(16\right)\\ x^2=32\\ x=\pm 4\sqrt{2}\left[\text{Taking square roots}\right]\end{array}$

Thus, the coordinates at which Juana live relative to the midpoint between the two towers are $\left(4\sqrt{2},6\right)$ and $\left(-4\sqrt{2},6\right)$ .