(a) A furniture manufacturer produces two types of display cabinets, type X and type Y. On a
weekly basis he must produce at least 2 of each type, but not more than 5 of type X or more
than6oftypeY.Ittakes4hourstoproducetypeXand5hoursfortypeYina40hourworking
week. At least 12 workers are needed with 2 working on type X and 3 on type Y at any one
time.
• Represent the information above as a system of inequalities 2
• Draw a graph of the system and indicate the feasible region clearly • Iftheprofit(P)ontypeXisTZS800andontypeYisTZS1000,writedowntheobjective
function in the form P = ax + by . • Determinethenumberofeachtypethatmustbeproducedeachweektomakeamaximum
profit. Determine the maximum profit.
(b) Each month a store owner can spend at most $100,000 on PC’s and laptops. A PC costs the
store owner $1000 and a laptop costs him $1500. Each PC is sold for a profit of $400 while
laptop is sold for a profit of $700. The store owner estimates that at least 15 PC’s but no more
than 80 are sold each month. He also estimates that the number of laptops sold is at most half
thePC’s. HowmanyPC’sandhowmanylaptopsshouldbesoldinordertomaximizetheprofit?
Question3 A 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 TZS 10,500 and TZS 9,000 respectively. To
control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10
litresperhectare. Further,nomorethan800litresofherbicideshouldbeusedinordertoprotect
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?
Answers
Answer:
a) A furniture manufacturer produces two types of display cabinets, type X and type Y. On a
weekly basis he must produce at least 2 of each type, but not more than 5 of type X or more
than6oftypeY.Ittakes4hourstoproducetypeXand5hoursfortypeYina40hourworking
week. At least 12 workers are needed with 2 working on type X and 3 on type Y at any one
time.
• Represent the information above as a system of inequalities 2
• Draw a graph of the system and indicate the feasible region clearly • Iftheprofit(P)ontypeXisTZS800andontypeYisTZS1000,writedowntheobjective
function in the form P = ax + by . • Determinethenumberofeachtypethatmustbeproducedeachweektomakeamaximum
profit. Determine the maximum profit.
(b) Each month a store owner can spend at most $100,000 on PC’s and laptops. A PC costs the
store owner $1000 and a laptop costs him $1500. Each PC is sold for a profit of $400 while
laptop is sold for a profit of $700. The store owner estimates that at least 15 PC’s but no more
than 80 are sold each month. He also estimates that the number of laptops sold is at most half
thePC’s. HowmanyPC’sandhowmanylaptopsshouldbesoldinordertomaximizetheprofit?
Question3 A 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 TZS 10,500 and TZS 9,000 respectively. To
control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10
litresperhectare. Further,nomorethan800litresofherbicideshouldbeusedinordertoprotect
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?
Answer:
The feasible region are:
A = (15, 15)
B = (80, 20)
C = (66 2/3, 33
Step-by-step explanation:
Let's represent the number of type X cabinets produced in a week by x, and the number of type Y cabinets by y. Then, the system of inequalities can be represented as:
2 ≤ x ≤ 5 (at least 2, but not more than 5 type X cabinets)
0 ≤ y ≤ 6 (not more than 6 type Y cabinets)
4x + 5y ≤ 40 (total hours for production should not exceed 40 hours)
2x + 3y ≥ 12 (at least 12 workers are needed, with 2 working on type X and 3 on type Y)
|
6 | A
| X
5 | /
| /
4 | B/
| /
3 | /
| /
2 | /
| Y/
1 | /
|/
----------
2 3 4 5 6
The feasible region is the shaded area between the lines AB, X, and Y, including the points A and B.
The objective function in the form P = ax + by is:
P = 800x + 1000y (profit on type X is TZS 800, profit on type Y is TZS 1000)
To determine the number of each type that must be produced to maximize profit, we need to find the coordinates of the vertices of the feasible region and evaluate the objective function at each vertex. The maximum profit will be the largest value obtained.
The vertices of the feasible region are:
A = (2, 4)
B = (5, 0)
C = (3, 3)
D = (2, 2)
Evaluating the objective function at each vertex, we get:
PA = 800(2) + 1000(4) = TZS 5200
PB = 800(5) + 1000(0) = TZS 4000
PC = 800(3) + 1000(3) = TZS 4800
PD = 800(2) + 1000(2) = TZS 4000
Therefore, the maximum profit is TZS 5200, which can be obtained by producing 2 type X cabinets and 4 type Y cabinets.
Let's represent the number of PCs sold each month by x, and the number of laptops sold by y. Then, the system of inequalities can be represented as:
x + y ≤ 100 (total spending should not exceed $100,000)
1000x + 1500y ≤ 100000 (total cost should not exceed $100,000)
15 ≤ x ≤ 80 (at least 15, but no more than 80 PCs)
y ≤ x/2 (number of laptops sold is at most half the number of PCs sold)
The objective function in the form P = ax + by is:
P = 400x + 700y (profit on each PC is $400, profit on each laptop is $700)
To determine the number of PCs and laptops that should be sold to maximize profit, we need to find the coordinates of the vertices of the feasible region and evaluate the objective function at each vertex. The maximum profit will be the largest value obtained.
The vertices of the feasible region are:
A = (15, 15)
B = (80, 20)
C = (66 2/3, 33
Here's the graph of the system of inequalities:
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