Chapter 8.3, Problem 68PS

Solution

(a)

To graph : The given situation using circles in a coordinate plane.

Given information :

The Modified Mercalli Intensity Scale rates an earthquake's "shaking strength". In general, the rating decreases as distance from the earthquake's epicenter increases. Suppose an earthquake has a Mercalli rating of $6.0$ at its epicenter, a $5.7$ rating $15$ miles away from the epicenter, a $5.4$ rating $25$ miles away and a $5.1$ rating $35$ miles away.

Graph :

The graph representing the given situation will be as below:

  

Interpretation :

The distance from the epicenter will be the radius of the circles.

(b)

To explain : An inequality for each circle describing the coordinates of locations with Mercalli rating at least as great as the Mercalli rating represented by the circle.

The three required equations are $x^2+y^2\le {15}^2$ , $x^2+y^2\le {25}^2$ , and $x^2+y^2\le {35}^2$ .

Given information :

The Modified Mercalli Intensity Scale rates an earthquake's "shaking strength". In general, the rating decreases as distance from the earthquake's epicenter increases. Suppose an earthquake has a Mercalli rating of $6.0$ at its epicenter, a $5.7$ rating $15$ miles away from the epicenter, a $5.4$ rating $25$ miles away and a $5.1$ rating $35$ miles away.

Explanation :

For the first earthquake having Mercalli rating $\left(R\right)$

  $5.7$ , which is $15$ miles away from the epicenter, the inequality will be,

  $5.7\le R\le 6$ , for $x^2+y^2\le {15}^2$ ......… (1)

For the second earthquake having Mercalli rating $\left(R\right)$

  $5.4$ , which is $25$ miles away from the epicenter, the inequality will be,

  $5.4\le R\le 5.7$ , for $x^2+y^2\le {25}^2$ ......… (2)

For the third earthquake having Mercalli rating $\left(R\right)$

  $5.1$ , which is $35$ miles away from the epicenter, the inequality will be,

  $5.1\le R\le 5.4$ , for $x^2+y^2\le {35}^2$ ......… (3)

Therefore, the three required equations are $x^2+y^2\le {15}^2$ , $x^2+y^2\le {25}^2$ , and $x^2+y^2\le {35}^2$ .

(c)

To calculate : The Mercalli rating $12$ miles west and $16$ miles south of the epicenter.

The prediction can be made using $x^2+y^2\le {25}^2$ .

Given information :

The Modified Mercalli Intensity Scale rates an earthquake's "shaking strength". In general, the rating decreases as distance from the earthquake's epicenter increases. Suppose an earthquake has a Mercalli rating of $6.0$ at its epicenter, a $5.7$ rating $15$ miles away from the epicenter, a $5.4$ rating $25$ miles away and a $5.1$ rating $35$ miles away.

Explanation :

The value of Mercalli rating $12$ miles west and $16$ miles south of the epicenter is between $15\text{ and }25$ miles, so the inequality $x^2+y^2\le {25}^2$ will be the most appropriate for making any prediction.

Therefore, the prediction can be made using $x^2+y^2\le {25}^2$ .