A manufacturer produces two models of toy airplanes. It takes the manufacturer 32 minutes to assemble model A and 8 minutes to package it. It takes the manufacturer 20 minutes to assemble model B and 10 minutes to package it. In a given week, the total available time for assembling is 3200 minutes, and the total available time for packaging is 960 minutes. Model A earns a profit of $10 for each unit sold and model B earns a profit of $8 for each unit sold. Assuming the manufacturer is able to sell as many models as it makes, how many units of each model should be produced to maximize the profit for the given week? Note that the ALEKS graphing calculator can be used to make computations easier.

The manufacturer should produce; Model A = 80 units, and Model B = 32 units.

Step-by-step explanation: The available time for assembly and packaging of both models have been provided as follows;

Total assembly time = 3200 minutes

Total packaging time = 960 minutes

From the information given, model A requires 32 minutes to assemble and model B requires 20 minutes to assemble. We can express this as an equation as follows;

32A + 20B = 3200 ———(1)

Also model A requires 8 minutes to package and model B required 10 minutes to package. This too can be expressed as follows;

8A + 10B = 960 ———(2)

We now have a pair of simultaneous equations which we shall solve using the elimination method. Multiply equation equation (1) by 10 and multiply equation (2) by 20 and we now have;

320A + 200B = 32000 ———(3)

160A + 200B = 19200 ———(4)

Subtract equation (4) from equation (3)

160A = 12800

Divide both sides of the equation by 160

A = 80

We can now substitute for the value of A into equation (1)

32(80) + 20B = 3200

2560 + 20B = 3200

Subtract 2560 from both sides of the equation

20B = 640

Divide both sides of the equation by 20

B = 32

Therefore in order to maximize profit for the given week, the manufacturer should produce 80 units of model A and 32 units of model B. This would yield a total value of;

(Model A) 80 x 10 = $800

(Model B) 32 x 8 = $256

Total sales for the week = 800 + 256

Total sales for the week = $1,056

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