Question

Carlo and Anita make mailboxes and toys in their craft shop near Lincoln. Each mailbox requires 1 hour of work from Carlo and 2 hours from Anita. Each toy requires

1 hour of work from Carlo and 1 hour from Anita. Carlo cannot work more than 5 hours per week and Anita cannot work more than 8 hours per week. If each mailbox sells for $8 and each toy sells for $12?, then how many of each should they make to maximize their? revenue? What is their maximum? revenue?

Carlo and Anita should make _mailboxes and _toys. Their maximum revenue is _?$.

Answer 1

Let 'x' denotes mailboxes and 'y' denotes toys.

Objective function is maximisation of revenue:

$ 8x + $ 12 y

Subject to two contraints:

x+ y <= 5 (because Carlo is restricted by 5 hours per week).

2x + y <= 8 (because Anita is restricted by 8 hours per week).

For plotting Carlo's constraint : y= 5- x

When x=0, y=5

When y=0, x=5

For plotting Anita's constraint : y= 8 -2x

When x=0, y=8

When y=0, x=4

And at intersection

5-x = 8- 2x

We get , x=3

y=5-x = 5-3 = 2

Hence, at intersection , x=3 and y=2.

By plotting these we get the following graph:

Because of the time restrictions , the points are availbale for maximisation of revenue are the inner points:

(x1, y1) = (0,5) . At this, revenue = 8(0)+ 12(5)= $60

(x2 , y2) = (4,0) . At this, Revenue= 8(4) + 12(0)= $32

(x3, y3) = (3,2) . At this Revenue = 8(3) + 12(2) = 24 + 24 = $ 48.

So, we can see that the revenue is maximised at (0,5) i.e $60.

Hence, optimal solution is Carlo and Anita should make x= 0 mail boxes and y=5 toys.

Their maximum revenue = $60.