Math Answers

Evaluate the following line integrals along with the curve C.

(a) ∫

C (x² − 2y² ) ds; C is the line segment parametrized by r(t) = < t √ 2 , t √ 2> for 0 ≤ t ≤ 4.

A bag contains three of black ball, four green balls and five red balls. Three balls are drawn

from the bag without replacement. Use a tree diagram to find the probability that the balls

are all of different colours.

Determine whether the following vector fields are conservative on R 2 . If the vector field is conservative then find a potential function. F =<e^x cos y, −e^x sin y>.

7. A manufacturer has three machines I, II and III installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas machine III must be operated for at least 5 hours a day. She produces only two items M and N each requiring the use of all the three machines. The number of hours required for producing 1unit of each of M & N on the three machines are given in following table

Items Numbers of hours required on machines

I II III

M 1 2 1

N 2 1 1.25

She makes a profit of Birr 600 and Birr 400 on items M and N respectively. How many

of each item should she produce so as to maximize her profit assuming that she can sell

all the items that she produced? What will be the maximum profit? (Solve through simplex method)

3. Tigist wishes to mix two types of food C and D in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin C and 11 units of vitamin D. Food C costs $ 60/kg and Food D costs $80/kg. Food C contains 3 units/kg of Vitamin A and 5 units / kg of Vitamin B while food D contains 4 units/kg of Vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture by graphic model.

2. A furniture manufacturer makes two products - tables and chairs. Processing of these products is done on two types of machines A and B. A chair requires 2 hours on machine type A and 6 hours on machine type B. A table requires 5 hours on machine type I and no time on Machine type II. There are 16 hours/day available on machine type A and 30 hours/day on machine type B. Profits gained by the manufacturer from a chair & a table are Birr 2 and Birr 10 respectively.

A. What should be the daily production of each of the two products?

B. Use graphical method of LPP to find the solution.

11. Wodera cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated as Birr 10,500 and Birr 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 liters and 10 liters per hectare. Further, no more than 800 liters of herbicide should be used in order to protect fish and wild life using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximize the total profit of the society? (Solve By graphic method).

9. A company manufactures two products P1 and P2. Profit per unit for P1 is $200 and for

P2 is $300. Three raw materials M1, M2 and M3 are required. One unit of P1 needs 5 units of M1 and 10 units of M2. One unit of P2 needs 18 units of M2 and 10 units of M3. Availability is 50 units of M1, 90 units of M2 and 50 units of M3.

A. Formulate as LPP

B. Find the optimal by using simplex method

graph of y = x³ + 6 is translated 4 units to the right.

The translated graph has equation y = f(x)

Work out f(x).

Give your answer in the form x³+ ax² + bx + c where a, b and c are integers.

6. There are two types of fertilizers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 costs Birr 6/kg and F2 costs Birr 5/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost? (solve through simplex method)