Question

A manufacturer of two products (Widgets and Gadgets) makes a profit of $110 for each widget sold and $100 for each gadget sold. However, production of these products generates hazardous waste charges at the rate of 1W² (where W is the number of Widgets produced) and 3G² (where G is the number of Gadgets produced). The manufacturer has ample supplies of all raw materials and can sell all the Widgets and Gadgets it produces. The firm has sufficient capacity to produce any realistic quantity of both products.

What is the optimal quantity of each product to produce? (Round your answers to 1 decimal place.)

_______ Widgets and ______ Gadgets

What is the maximum profit the manufacturer can earn? (Round your answer to 2 decimal places.)

Maximum Profit _________

Answer 1

Amounts are in $

Given the profit on W is $110 and the profit on G is $100

And the additional cost on Production of W generates additional cost of 1W^2 (W is number of W produced) and Production of G generates additional cost of 3G^2 (G is number of G produced)

In order to maximise profit, we have to produce goods untill the marginal profit is higher than marginal cost

That means until Derivative of 1W^2 is lesser than 110 and Derivate of 3G^2 is lesser than 100

So

Derivative of 1W^2 is less than equal to 110

1x2xW is less than or equal to 110 [d/dx 1W^2 is 1x2xW]

W is less than or equal to 110/2

W is less than or equal to 55

At this number of units, we get maximum profit

For Gadget

Derivate of 3(G)^2 less than or equal to 100

3x2xG is less than or equal to 100

G is less than or equal to 100/6

G is less than or equal to 17 (rounded off)

At this number of units, the profit is maximum

Maximum amount of profit will be

= No of units of Widgets x profit for widget - 1(W)^2 + No of units of Gadgets x Profit for gadget - 3(G)^2

= 55 x 110 - 1(55)^2 + 17 x 100 - 3(17)^2

= 6,050 - 3,025 + 1,700 - 867

= 3,858

So for the first two blanks the answers will be 55 widgets and 17 gadgets and for the 3rd blank of maximum profit the answer is 3,858