Chapter 9.8, Problem 37PPS

(a)

To determine

To fill the blank space in the box with the accurate value.

Answer to Problem 37PPS

The required size is 2.375 yards.

Explanation of Solution

Given information:

Rate of change is the change in $y$ divided by the change in $x$ .

The given table is

Richer Number x	Increase in magnitude y	Rate of change slop
1	1
2	10	9
3	100
4	1000
5	10000

Let Richter number is $x$ and increased in magnitude is $y$ .

Now, find the rate of change. Consider the first two row.

  $\begin{array}{c}\text{Rate}=\frac{10-1}{2-1}\\ =9\end{array}$

Similarly, consider second and third row.

  $\begin{array}{c}\text{Rate}=\frac{100-10}{3-2}\\ =90\end{array}$

Similarly find all the rate of change for different values of $x\text{and}y$ .

Hence, the required table is

Richter Number $x$	Increase in magnitude $y$	Rate of change
1	1
2	10	9
3	100	90
4	1000	900
5	10000	9000

(b)

To determine

To graph: The points in the Richter vs magnitude plane.

Explanation of Solution

Given information:

The table for the rate of change is

Richter Number $x$	Increase in magnitude $y$	Rate of change
1	1
2	10	9
3	100	90
4	1000	900
5	10000	9000

Graph:

Consider Richter number as $x$ coordinate and magnitude as $y$ coordinates.

  

Interpretation:

Hence, from the graph the $\left(x,y\right)$ points are $\left(1,1\right),\left(2,10\right),\left(3,100\right),\left(4,1000\right)\text{and}\left(5,10000\right)$ .

(d)

To determine

The exponential equation that represent the Richter scale.

Answer to Problem 37PPS

The required equation is $x^2=0$ .

Explanation of Solution

Given information:

The table for the rate of change is

Richter Number $x$	Increase in magnitude $y$	Rate of change
1	1
2	10	9
3	100	90
4	1000	900
5	10000	9000

Formula Used:

Common ratio for a geometric sequence is $r=\frac{a_n}{a_{n-1}}$ .

And the n^th term is $a_n=a_1{r}^{n-1}$

Here $a_1$ is the first term, $a_n$ is the n^thterm and $r$ is the common ratio.

The given magnitude number are 1, 10, 100, 1000, 10000.

It is observed that the Richter number are in geometric sequence.

Thus, find the common ratio. Use the formula $r=\frac{a_n}{a_{n-1}}$

  $\frac{10}{1}=10$

Let for $x$ Richter number $y$ be the magnitude.

Thus, find $y^{\text{th}}$ term. Use the formula $a_n=a_1{r}^{n-1}$ .

  $\begin{array}{c}y=1\times {10}^{x-1}\\ y={10}^{x-1}\end{array}$