Question

The manufacturing company manufactures large and small book shelves. Large shelves require 40 pounds of metal to fabricate and small shelves require 30 pounds, but the company has only 400 pounds of metal on hand. If the company can sell each large shelve for $80 and each small shelve for $49, how many each shelve should it manufacture in order to maximize income?

Answer 1

We know that

• Large shelves require 40 pounds.

,

• Small shelves require 30 pounds.

,

• The company has only 400 pounds of metal.

,

• The selling price of each large shelf is $80.

,

• The selling price of each small shelf is $49.

First, let's make a table with the given data of the problem, this will help you organize it.

From the given information, we can define the following constraints.

$\begin{gathered} 40x+30y\leq400 \\ y\ge0 \\ x\ge0 \end{gathered}$

These constraints form the following region:

To find the maximum income possible, we have to evaluate the income function at (0,13.333) and (10,0).

$I(x,y)=80x+49y$

Observe that we formed the income function using the table above. Now, let's evaluate it at each point.

$\begin{gathered} I(0,13.333)=80\cdot0+49\cdot13.333=0+653.317=653.32 \\ I(10,0)=80\cdot10+0=800+0=800 \end{gathered}$

Therefore, they would have to sell 10 large shelves to reach a maximum income of $800.