A manufacturer of two products Widgets and Gadgets makes a p

A manufacturer of two products (Widgets and Gadgets) makes a profit of $180 for each widget sold and $180 for each gadget sold. However, production of these products generates hazardous waste charges at the rate of 3W² (where W is the number of Widgets produced) and 2G² (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.)

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

Solution

Q1)

As per the condition,

3W^2 = $180

W^2 = 180/3

W^2 = 60

W = Square root of 60

    = 7.7 units (Answer)

And,

2G^2 = $180

G^2 = 180/2

G^2 = 90

G = Square root of 90

    = 9.5 units (Answer)

Q2)

Profit for W = Per unit profit × Number of W

                           = $180 × 7.7

                           = $1,386

Profit for G = Per unit profit × Number of G

                           = $180 × 9.5

                           = $1,710

Maximum profit = 1,386 + 1,710

                            = $3,096 (Answer)