Chapter 4.4, Problem 8PPS

a.

To determine

The equation for the line of fit in slope-intercept form using the points $\left(2,21.75\right)\text{ and }\left(4,21\right)$ .

Answer to Problem 8PPS

The equation in slope-intercept form is $y=-0.375x+22.5$ .

Explanation of Solution

Given: The given points are $\left(2,21.75\right)\text{ and }\left(4,21\right)$ through which the line passes.

Calculation: The slope for the line of fit using the points $\left(2,21.75\right)\text{ and }\left(4,21\right)$ is evaluated as,

  $\begin{array}{c}m=\frac{21-21.75}{4-2}\mathrm{......}\left(\begin{array}{c}\left(x_1,y_1\right)\to \left(2,21.75\right)\\ \left(x_2,y_2\right)\to \left(4,21\right)\end{array}\right)\\ =-\frac{0.75}{2}\\ =-0.375\end{array}$

Now, the equation for the line of fit using the slope $\left(m=-0.375\right)$ and point $\left(4,21\right)$ is evaluated as,

  $\begin{array}{c}y-\left(21\right)=\left(-0.375\right)\left(x-\left(4\right)\right)\mathrm{......}\left(\begin{array}{l}\text{Here }m=-0.375\text{ and}\\ \left(x_1,y_1\right)\to \left(4,21\right)\end{array}\right)\\ y-21=-0.375x+1.5\\ y=-0.375x+\mathrm{22.5......}\left(1\right)\end{array}$

b.

To determine

The prediction for the milk production in the year 2015.

Answer to Problem 8PPS

The estimated milk production in the year 2015 is 16.875 gallons.

Explanation of Solution

Given: The given points are $\left(2,21.75\right)\text{ and }\left(4,21\right)$ through which the line of fit passes.

Calculation: The prediction for the milk production in the year 2015 can be evaluated by plugging $\left(x=2015-2000=15\right)$ in (1). Therefore, from (1),

  $\begin{array}{c}y=-0.375\left(15\right)+\mathrm{22.5......}\left(\text{For, }x=15\right)\\ =16.875\text{ gal}\end{array}$

c.

To determine

The year in which the consumption of milk will be 10 gallons.

Answer to Problem 8PPS

The equation for the line of fit is $y=0.11x-194.76$ .

Explanation of Solution

Given: The given points are $\left(2,21.75\right)\text{ and }\left(4,21\right)$ through which the line of fit passes.

Calculation: For, the consumption of milk to be 10 gallons, plugging $\left(y=10\right)$ in (1). Therefore, from (1),

  $\begin{array}{c}\left(10\right)=-0.375x+\mathrm{22.5......}\left(\text{For, }y=10\right)\\ \text{or, }0.375x=12.5\\ x=33.33\end{array}$

Hence, adding 33 to the base year 2000, the year we get is 2033. Therefore, in the year 2033 the consumption of milk will be 10 gallons.

d.

To determine

Can the equation of line of fit give the reasonable estimate for any year to find the consumption of milk? Explain.

Answer to Problem 8PPS

Yes, the equation for the line of fit gives reasonable estimate for any year to find the consumption of milk.

Explanation of Solution

Given: The given points are $\left(2,21.75\right)\text{ and }\left(4,21\right)$ through which the line of fit passes.

Calculation: Yes, the equation for the line of fit gives reasonable estimate for any year to find the consumption of milk because the equation for the line of fit represents the close approximation trend between the given two variables.