Chapter 8.5, Problem 31E

a.

To determine

To write:

An equation giving the unharvested area y of the larger field (in acres) after x days.

Answer to Problem 31E

The equation $y=1000-50x$ represents the unharvested area y of the larger field (in acres) after x days.

Explanation of Solution

Given:

Two farmers each harvest 50 acres of corn per day from their fields. The area of one farmer’s field is 1000 acres, and the area of the other farmer’s field is 600 acres.

Calculation:

Let x represent number of days and y represent unharvested area.

We have been given that the farmer harvest 50 acres of corn per day from their fields, so number of acres harvested in x days would be $50x$ .

The unharvested area after xdays would be total area minus area harvested in x days. The larger area is 1000 acres.

  $y=1000-50x$

Therefore, the equation $y=1000-50x$ represents the unharvested areaof the larger field (in acres) after x days.

b.

To determine

To write:

An equation giving the unharvested area y of the smaller field (in acres) after x days.

Answer to Problem 31E

The equation $y=600-50x$ represents the unharvested area of the smaller field (in acres) after x days.

Explanation of Solution

Given:

Two farmers each harvest 50 acres of corn per day from their fields. The area of one farmer’s field is 1000 acres, and the area of the other farmer’s field is 600 acres.

Calculation:

Let x represent number of days and y represent unharvested area.

We have been given that the farmer harvest 50 acres of corn per day from their fields, so number of acres harvested in x days would be $50x$ .

The unharvested area after x days would be total area minus area harvested in x days. The smaller area is 600 acres.

  $y=1000-50x$

Therefore, the equation $y=600-50x$ represents the unharvested area of the smaller field (in acres) after x days.

c.

To determine

To graph:

The equations from part (a) and part (b) in the same coordinate plane. Identify the slope and y -intercept of each graph.

Answer to Problem 31E

Our required graph would look like:

  

Explanation of Solution

Given:

Two farmers each harvest 50 acres of corn per day from their fields. The area of one farmer’s field is 1000 acres, and the area of the other farmer’s field is 600 acres.

Concept used:

The equation of a line in slope-intercept form is $y=mx+b$ , where m represents slope of line and b represents the y -intercept.

Calculation:

Upon comparing equation $y=1000-50x$ with slope-intercept form, we can see that slope is $-50$ and the y -intercept is 1000.

Upon comparing equation $y=600-50x$ with slope-intercept form, we can see that slope is $-50$ and the y -intercept is 600.

Upon graphing our both equations on same coordinate plane, we will get our required graph as shown below:

  

d.

To determine

What is the geometric relationship between graphs from part (c)? How do you know?

Answer to Problem 31E

The area under the graph of $y=1000-50x$ is more than the area under the graph of $y=600-50x$ .

Explanation of Solution

Given:

Two farmers each harvest 50 acres of corn per day from their fields. The area of one farmer’s field is 1000 acres, and the area of the other farmer’s field is 600 acres.

Calculation:

Upon looking at our graph in part (c), we can see that the graph of $y=1000-50x$ covers more area than the graph of $y=600-50x$ .

Therefore, the area under the graph of $y=1000-50x$ is more than the area under the graph of $y=600-50x$ .

e.

To determine

How long does it take to harvest the corn in the larger field? in the smaller field?

Answer to Problem 31E

It will take 20 days to harvest the corn in the larger field.

It will take 12 days to harvest the corn in the smaller field.

Explanation of Solution

Given:

Two farmers each harvest 50 acres of corn per day from their fields. The area of one farmer’s field is 1000 acres, and the area of the other farmer’s field is 600 acres.

Calculation:

To find the number of days, it will take to harvest the corn in the larger field, we will substitute $y=0$ in equation $y=1000-50x$ and solve for x as:

  $\begin{array}{l}y=1000-50x\\ 0=1000-50x\\ 50x=1000\left(\text{Adding 50}x\text{ on both sides}\right)\\ \frac{50x}{50x}=\frac{1000}{50}\left(\text{Dividing both sides by 50}\right)\\ x=20\end{array}$

Therefore, it will take 20 days to harvest the corn in larger field.

To find the number of days, it will take to harvest the corn in the smaller field, we will substitute $y=0$ in equation $y=600-50x$ and solve for x as:

  $\begin{array}{l}y=600-50x\\ 0=600-50x\\ 50x=600\left(\text{Adding 50}x\text{ on both sides}\right)\\ \frac{50x}{50x}=\frac{600}{50}\left(\text{Dividing both sides by 50}\right)\\ x=12\end{array}$

Therefore, it will take 12 days to harvest the corn in smaller field.