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}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%2040x%2B30y%5Cleq400%20%5C%5C%20y%5Cge0%20%5C%5C%20x%5Cge0%20%5Cend%7Bgathered%7D)
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](https://tex.z-dn.net/?f=I%28x%2Cy%29%3D80x%2B49y)
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}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20I%280%2C13.333%29%3D80%5Ccdot0%2B49%5Ccdot13.333%3D0%2B653.317%3D653.32%20%5C%5C%20I%2810%2C0%29%3D80%5Ccdot10%2B0%3D800%2B0%3D800%20%5Cend%7Bgathered%7D)
<h2>Therefore, they would have to sell 10 large shelves to reach a maximum income of $800.</h2>