You manage a crew of 160 workers who could be assigned to make either of two pro

Question

You manage a crew of 160 workers who could be assigned to

make either of two products. Product A requires 2 workers per unit of

output. Product B requires 4 workers per unit of output.

(a) Write an equation to express the combinations of products A and

B that could be produced using exactly 160 workers. 2A + 4B =

160.

(b) Suppose now that every unit of product A that is produced requires

the use of 4 shovels as well as 2 workers and that every unit of product B

produced requires 2 shovels and 4 workers. On the graph you have just

drawn, use red ink to shade in the area depicting combinations of A and

B that could be produced with 180 shovels if there were no worries about

the labor supply. Write down an equation for the set of combinations of

A and B that require exactly 180 shovels. 4A + 2B = 180.

(c) On the same diagram, use black ink to shade the area that represents

possible output combinations when one takes into account both the

limited supply of labor and the limited supply of shovels.

(d) On your diagram locate the feasible combination of inputs that use up

all of the labor and all of the shovels. If you didn’t have the graph, what

equations would you solve to determine this point? 2A + 4B =

160 and 4A + 2B = 180.

(e) If you have 160 workers and 180 shovels, what is the largest amount of

product A that you could produce? 45 units. If you produce this

amount, you will not use your entire supply of one of the inputs. Which

one? Workers. How many will be left unused? 70.

Please explain the full steps to reach the answers to part e. Answers are in bold letters

Explanation / Answer

E) We know that 2 workers can produce 1 unit of A, so it means that we can make 160 / 2 = 80 units of A. But here we also need shovels, which are used 4 per unit of A produced. Shovels are in a limited supply so we employ all the 180 shovels we can produce 180 / 4 = 45 units.

So we cannot increase the production of a beyond 45 units due to the constraints of shovels.

technically we can say that for making one unit of A we need 2L and 4S where L is workers and S is shovels.

A = 1/2L + 1/4S, subject to the condition that S cannot go beyond 180

Hence we will make only 45 units. At the level of 45 units we will only need 45* 2 = 90 units of workers. Thus 160 - 90 = 70 units of workers will be left unutilized.

Units produced = 45 units, Workers left unused will be 70.

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