Chapter 9.1, Problem 41E

(a)

To determine

To find: The standard equation that describes the outer boundary of the earthquake if the epicenter is at $\left(0,0\right)$ .

Answer to Problem 41E

The standard equation that describes the outer boundary of the earthquake is $x^2+y^2={52}^2$ .

Explanation of Solution

Given information:

An earthquake was felt up to $52$ miles from its epicenter. You were located $40$ miles west and $30$ miles south of the epicenter. The epicenter be at the point $\left(0,0\right)$ .

Formula Used:

Use the general equation of circle with center at $\left(h,k\right)$ and radius $r$ .

  ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$

Calculation:

An earthquake was felt up to $52$ miles from its epicenter, this means that the distance from epicenter to the boundary is always constant. So, the boundary of earthquake is a circle. The center of the circle(epicenter) is located at the point $\left(0,0\right)$ .

The earthquake was felt up to $52$ miles from its epicenter means that the radius of the circle of the boundary of earthquake is $52$ miles.

Substitute $0$ , $0$ for $h$ , $k$ and $52$ for $r$ respectively in the general equation.

  $\begin{array}{c}{\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2\\ {\left(x-0\right)}^2+{\left(y-0\right)}^2={52}^2\\ x^2+y^2={52}^2\end{array}$

Therefore, the standard equation that describes the outer boundary of the earthquake is $x^2+y^2={52}^2$ .

(b)

To determine

To check: The earthquake was felt $40$ miles west and $30$ miles south of the epicenter or not.

Answer to Problem 41E

Yes, the earthquake was felt at the $40$ miles west and $30$ miles south of the epicenter.

Explanation of Solution

Given information:

An earthquake was felt up to $52$ miles from its epicenter. You were located $40$ miles west and $30$ miles south of the epicenter. The epicenter be at the point $\left(0,0\right)$ .

Calculation:

If the earthquake was felt at given location, then the distance of the distance of the given point from the epicenter must be less than or equal to the radius of the circle.

The location for $40$ miles west and $30$ miles south of the epicenter is represented by the point $\left(-40,-30\right)$ .

Use the distance formula between the two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ .

  $\text{Distance}=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Substitute $0$ , $0$ for $x_1$ , $y_1$ and $-40$ , $-30$ for $x_2$ , $y_2$ respectively in the distance formula.

  $\begin{array}{c}\text{Distance}=\sqrt{{\left(-40\right)}^2+{\left(-30\right)}^2}\\ =\sqrt{1600+900}\\ =\sqrt{2500}\\ =50\end{array}$

As the distance of the location from the epicenter is less than the radius of the circle representing the boundary of the earthquake.

Therefore, the earthquake was felt at the $40$ miles west and $30$ miles south of the epicenter.

(c)

To determine

To prove: The answer of part(b) by graphing the equation of the outer boundary of earthquake and the location and also calculate the distance of the location from the boundary of the earthquake.

Answer to Problem 41E

The graph of the equation of outer boundary of earthquake and location is plotted in Figure (1). The distance of the location from the outer boundary of the earthquake is $2$ miles.

Explanation of Solution

Given information:

An earthquake was felt up to $52$ miles from its epicenter. You were located $40$ miles west and $30$ miles south of the epicenter. The epicenter be at the point $\left(0,0\right)$ .

Calculation:

The graph of the equation of outer boundary of earthquake and location is shown below.

  

Figure (1)

As the point of location is lying inside the circle of the outer boundary of earthquake.

Therefore, the earthquake was felt at the $40$ miles west and $30$ miles south of the epicenter.

As calculated in part(b), the distance of the location from the epicenter is $50$ miles and the radius of the circle is $52$ miles.

Therefore, the distance of the location from the outer boundary of the earthquake is $2$ miles.