4. A rectangular stock pen is to be built using a total of 600 ft of fencing. Part of this fencing will be used to build a fence across the middle of the rectangle (see diagram). Find the length and width of the rectangle that give the maximum total area.
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Nellymegs I think I can visualize the diagram. Here are the facts that you have been given.
Fact 1. You have 600 ft. of fence
Fact 2. You are told to place a fence across the middle
Fact 3. You want maximum volume.
What is the total fencing used, we know it is 600 ft, but we need to lay it out!
The fence will be a perimeter for the main rectangle plus the section across the middle.
Let l = Length and let w = width of the main section. The section across the middle would also be the same as width. Remembering this the amount of fence is distributed like this:
600=3w + 2l Do you understand that equation?
Now the area is simply A=wl. This is all we need to figure the maximum volume.
From the first equation (600=3w+2l) Express l in terms of w.
Now use that in the Area equation like this
Differentiate and set to zero to solve for max volume.
300-3w=0 Solve for w by subtracting 300 from both sides
-3w=-300 divide both sides by -3
The area is 15,000 sq ft. Also I mistakenly referred to area as volume in the above post.