Chapter 3.1, Problem 51AYU

Developing a Linear Model from Data How many songs can an iPod hold? The following data represent the memory $m$ and the number of songs $n$.

a. Plot the ordered pairs $(m,n)$ in a Cartesian plane.

b. Show that the number of songs $n$ is a linear function of memory $m$.

c. Determine the linear function that describes the relation between $m$ and $n$.

d. What is the implied domain of the linear function?

e. Graph the linear function in the Cartesian plane drawn in part (a).

f. Interpret the slope.

To determine

To calculate:

a. Plot the ordered pairs $\left(m,n\right)$ in a Cartesian plane.

b. Show that the number of songs n is a linear function of memory $m$.

c. Determine the linear function that describes the relation between $m$ and $n$.

d. What is the implied domain of the linear function?

e. Graph the linear function in the Cartesian plane drawn in part (a..

f. Interpret the slope.

Answer to Problem 51AYU

Solution:

a. The plot of the data in Cartesian plane is as shown below:

b. Yes, it is proved that the number of songs n is a liner function of memory $m$.

c. The linear function of the given data is $y=218.75x$.

d. The implied domain of the given function is $(-\infty ,\infty )$.

e. The graph is as shown below:

f. The function is an increasing function.

Explanation of Solution

Given:

The given data is

Formula Used:

Average rate of change if a function $f(x)$ from $a$ to $b$ is $m=\frac{f(b)-f(a)}{b-a}$.

Point slope formula: $y-y_1=m(x-x_1)$

Calculation:

a. The plot of the given data in a Cartesian plane is

b. Here, in the above Cartesian plane, we can see that the points are arranged in a straight line format.

On joining the given points, we get a straight non vertical line.

Therefore, we can say that the given number of songs $n$ is a linear function of memory $m$.

c. From the given data, we can find the average rate of change.

When

$a=8$

$b=16$

$f(a)=1750$

$f(b)=3500$

Now, we get the average rate of change of the given function as

$m=\frac{3500-1750}{16-8}$

$=\frac{1750}{8}=218.75$

Thus, the rate of change of the given function is $218.75$.

Now, we have to use this in the point slope formula, where $x_1=8$ and $y_1=1750$.

Thus, we get

$y-y_1=m(x-x_1)$

$\Rightarrow y-1750=218.75(x-8)$

$\Rightarrow y=218.75x-1750+1750$

$\Rightarrow y=218.75x$

Therefore, the linear function of the given data is $y=218.75x$

d. We can see that the value of $x$ in the linear function can be any number. Therefore, the implied domain of the given function is $(-\infty ,\infty )$.

e. The graph of the linear function is

f. We know that the average rate of change of the function is itself its slope.

Thus, here we have seen that the slope is $218.75$ which is a positive number.

Therefore, the function is an increasing function.