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a.
To write:
An equation giving the unharvested area y of the larger field (in acres) after x days.
a.
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 write:
An equation giving the unharvested area y of the smaller field (in acres) after x days.
b.
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 graph:
The equations from part (a) and part (b) in the same coordinate plane. Identify the slope and y -intercept of each graph.
c.
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.
What is the geometric relationship between graphs from part (c)? How do you know?
d.
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.
How long does it take to harvest the corn in the larger field? in the smaller field?
e.
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:
y=1000−50x0=1000−50x50x=1000(Adding 50x on both sides)50x50x=100050(Dividing both sides by 50)x=20
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:
y=600−50x0=600−50x50x=600(Adding 50x on both sides)50x50x=60050(Dividing both sides by 50)x=12
Therefore, it will take 12 days to harvest the corn in smaller field.
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