Chapter 9.1, Problem 91E

(a)

To determine

To find the standard equation that describes the outer boundary of the earthquake.

Answer to Problem 91E

The standard equation that describes the outer boundary of the earthquake is

  $X^2+Y^2=2704$

Explanation of Solution

Given:

Earthquake is felt up to 52 miles from its epicentre. Let the epicentre be at point $\left(0,0\right)$

Concept Used:

Outer boundary of the earthquake will be in form of circle.

So,

Standard Equation of earthquake will be similar to Equation of circle

  ${\left(X-a\right)}^2-{\left(Y-b\right)}^2={\left(radius\right)}^2$

Calculation:

Given epicentre is at point $\left(0,0\right)$

Therefore,

  $\left(a,b\right)\to \left(0,0\right)$

Hence,

Standard equation that describes earthquake is as follows

  $\begin{array}{l}{\left(X-0\right)}^2+{\left(Y-0\right)}^2={\left(52\right)}^2\\ X^2+Y^2=2704\end{array}$

Conclusion:

  $X^2+Y^2=2704$ is required standard equation of epicentre.

(b)

To determine

To tell whether the earthquake be felt by the person who is located at 40 miles west and 30 miles south of the epicentre or not

Answer to Problem 91E

Yes, the earthquake be felt by the person who is located at 40 miles west and 30 miles south of the epicentre.

Explanation of Solution

Given:

Location of a person is 40 miles west and 30 miles south of the epicentre.

Concept Used:

Find distance between the person and epicentre. If distance is within boundary of earthquake then it will be felt.

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

Calculation:

Epicentre is at origin $\left(0,0\right)$ and coordinates of person are $\left(40,30\right)$

Distance

  $\begin{array}{l}=\sqrt{{\left(40-0\right)}^2+{\left(30-0\right)}^2}\\ =\sqrt{1600+900}\\ =\sqrt{2500}\\ =50\end{array}$

Conclusion:

As earthquake is felt up to 52 miles, therefore it will surely be felt by the person who is located at given coordinates.

(c)

To determine

To verify the answer of part (b) by graphing equation of outer boundary and plotting location of person and also find how far is person from boundary.

Answer to Problem 91E

Person is 2 miles away from the boundary

Explanation of Solution

Given:

Epicentre is 52 miles away from boundary of earthquake.

Concept Used:

Required distance =Distance between boundary and epicentre $-$ Distance between person and epicentre

Calculation:

Required graph

  

Conclusion:

Required distance = $52-50=2\text{ miles}$