Now, we’re ready to go back to the problem we started on the previous page.
Example 1: Planting Crops
A farmer has a certain amount of time to plant his crops. He is planting tomatoes and beans, but in order to harvest them in time to meet the contract he has made with a canning company, he must plant within 21 days. He can plant beans at a rate of 200 acres a day, and he can plant tomatoes at a rate of 150 acres a day. Altogether, he will use, at most 2,100 acres. Using the system below to represent his constraints, how many acres of each type of crop should he plant?
Let T represent the number of days planting tomatoes.
Let B represent the number of days planting beans.
We know that both T and B must be more than zero.
B + T ≤ 21
300B + 175T ≤ 2100
Start by graphing the system, substituting x for B and y for T,
x + y ≤ 21 and 300x + 175y ≤ 2100
Now, solve each in terms of y and graph.
y ≤ 21 – x and y ≤ 12 – 12/7 x
The solutions are in the dark green area and along the portions of the lines forming that area. The ordered pairs represent the number of days to plant each.
For example, if you choose the ordered pair (4, 5) in the solution area:
x = 4 days planting beans and y = 5 days planting tomatoes.
4(300) = 1200 acres of beans
5(175) = 875 acres of tomatoes
The total acreage planted is 2075, which is less than the 2,100 available.
Example 2: Making Punch
Ariel and Sarah are making punch for a party using a combination of orange sherbet and a lemon lime soda. They know from experience that they must use more than twice as many quarts of soda as quarts of sherbet, but not more than three times as much soda as orange sherbet. The total amount of punch should be between 10 and 12 quarts. Use the inequalities for this system listed below to find out how much soda and sherbet they should buy?
Let x = quarts of sherbet
Let y = quarts of soda
y ≤ 3x
x + y ≥ 10
x + y ≤ 12
There is a small quadrilateral in the center of the graph, where all four inequalities overlap. The ordered pairs with whole number coefficients within it include:
(3, 7), (3, 8), (3, 9), and (4, 8).
So they could buy any of these combinations and stay within the constraints:
Quarts sherbet
3
3
3
4
Quarts soda
7
8
9
8
Problem Solving 2 (7 points)
Find the 6- Problem Solving 2 link. You may do this more than one time to improve your score.
Quiz 3 (7 points)
It’s time to complete your homework. Find the 6-Section Homework link to submit it for a grade. You may take this only one time, so check your understanding of the material before completing your homework, and do your best.
Problem Solving 2 (
Quiz 3 (