Explanation Given information: A farmer wants to plant cauliflower and cabbage in 300 acres of arable land. He has $ 17500 for planting and $ 12000 for fertilizer. The cost of cauliflower to plant is $ 70 for 1 acre and cost of cabbage to plant is $ 35 .The cost of fertilizers is $ 25 and $ 55 for 1 acre of cauliflower and for 1 acre of cabbage respectively. Calculation: Consider x as the number of acre of cauliflower and y as the number of acre of cabbage. The total area of land with the farmer is 300 . Therefore, the inequality is x + y ≤ 300 . The cost of planting cauliflower is $ 70 for 1 acre and cost of planting cabbage is $ 35 and the total amount with the farmer is $ 17500 for planting, it means he can spend at most $ 17500 on planting. The inequality can be written as, 70 x + 35 y ≤ 17500 Fertilizer costs $ 25 for 1 acre of cauliflower and $ 55 for 1 acre of cabbage and the total amount with the farmer is $ 12000 to buy the fertilizers it means he can spend at most $ 12000 on fertilizers. The inequality can be written as, 25 x + 55 y ≤ 12000 Now, x and y represents the number of acre land. Therefore, x and y cannot be negative The inequality can be written as, x ≥ 0 , y ≥ 0 Therefore, the system of inequalities is, { x + y ≤ 300 70 x + 35 y ≤ 17500 25 x + 55 y ≤ 12000 x ≥ 0 , y ≥ 0 Graph: The system of inequalities is, x + y ≤ 300 ....... (1) 70 x + 35 y ≤ 17500 ....... (2) 25 x + 55 y ≤ 12000 ....... (3) The corresponding equation of the inequality (1) is, x + y = 300 Since, the inequality x + y ≤ 300 involves the less than or equal to sign, then the points on the line x + y = 300 satisfies the inequality x + y ≤ 300 . Therefore, the line x + y = 300 itself is the part of the solution set. The corresponding equation of the inequality (2) is, 70 x + 35 y = 17500 Since, the inequality 70 x + 35 y ≤ 17500 involves the less than or equal to sign, then the points on the line 70 x + 35 y = 17500 satisfies the inequality 70 x + 35 y ≤ 17500 . Therefore, the line 70 x + 35 y = 17500 itself is the part of the solution set. The corresponding equation of the inequality (3) is, 25 x + 55 y = 12000 Since, the inequality 25 x + 55 y ≤ 12000 involves the less than or equal to sign, then the points on the line 25 x + 55 y = 12000 satisfies the inequality 25 x + 55 y ≤ 12000 . Therefore, the line 25 x + 55 y = 12000 itself is the part of the solution set. Consider the test point ( 100 , 100 ) to check whether the solution satisfies each inequality of the given system or not. Substitute 100 for x and 100 for y in the inequality x + y ≤ 300 . x + y ≤ ? 300 100 + 100 ≤ ? 300 200 ≤ 300 The point ( 100 , 100 ) lies on the left side of the line x + y = 300 and satisfies the inequality x + y ≤ 300 . So it is the part of the solution set. Substitute 100 for x and 100 for y in the inequality 70 x + 35 y ≤ 17500 . 70 x + 35 y ≤ ? 17500 70 ( 100 ) + 35 ( 100 ) ≤ ? 17500 7000 + 3500 ≤ ? 17500 10500 ≤ 17500 The point ( 100 , 100 ) lies on the left side of the line 70 x + 35 y = 17500 and satisfies the inequality 70 x + 35 y ≤ 17500 . So it is the part of the solution set. Substitute 100 for x and 100 for y in the inequality 25 x + 55 y ≤ 12000 . 25 x + 55 y ≤ ? 12000 25 ( 100 ) + 55 ( 100 ) ≤ ? 12000 2500 + 5500 ≤ ? 12000 8000 ≤ 12000 The point ( 100 , 100 ) lies below the line 25 x + 55 y = 12000 and satisfies the inequality 25 x + 55 y ≤ 12000 To determine (b) To check: Whether 155 acres of cauliflower and 115 acres of cabbage can be planted by farmer or not. To determine (c) To check: Whether 115 acres of cauliflower and 175 acres of cabbage can be planted by farmer or not.

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 5.5, Problem 70E

To determine

(a)

To find:

The system of inequalities that describes the area of land that the farmer can plant with the available resources for each crop and graph the feasible region.

To determine

(b)

To check:

Whether 155 acres of cauliflower and 115 acres of cabbage can be planted by farmer or not.

To determine

(c)

To check:

Whether 115 acres of cauliflower and 175 acres of cabbage can be planted by farmer or not.

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Chapter 5.5, Problem 69E

Chapter 5.5, Problem 71E

Chapter 5 Solutions

College Algebra

Ch. 5.1 - The system of equations {2x+3y=75xy=9} is a system...Ch. 5.1 - A system of equation in two variable can be solved...Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Elimination method Use the elimination method to...Ch. 5.1 - Elimination method Use the elimination method to...Ch. 5.1 - Elimination method Use the elimination method to...Ch. 5.1 - Elimination method Use the elimination method to...Ch. 5.1 - Elimination method Use the elimination method to...Ch. 5.1 - Elimination method Use the elimination method to...

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(a) To find: The system of inequalities that describes the area of land that the farmer can plant with the available resources for each crop and graph the feasible region.

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(a) To find: The system of inequalities that describes the area of land that the farmer can plant with the available resources for each crop and graph the feasible region.

(b) To check: Whether 155 acres of cauliflower and 115 acres of cabbage can be planted by farmer or not.

(c) To check: Whether 115 acres of cauliflower and 175 acres of cabbage can be planted by farmer or not.

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Chapter 5 Solutions

(a)

To find:

The system of inequalities that describes the area of land that the farmer can plant with the available resources for each crop and graph the feasible region.

(b)

To check:

Whether 155 acres of cauliflower and 115 acres of cabbage can be planted by farmer or not.

(c)

To check:

Whether 115 acres of cauliflower and 175 acres of cabbage can be planted by farmer or not.