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5-39.

YELLOW SUBMARINE, Part One Ring O. Star is sick and tired of chasing blue meanies around the inside of his yellow submarine, so he decides to build a special detention chamber to hold the ones he catches. The first design he thinks of is a rectangular chamber with sides made of razor-sharp barbed wire. To save money, he decides to build the chamber up against a submarine wall—that way only three sides needed to be made out of barbed wire. He estimates the detention chamber needs to have 40 square meters of area. What dimensions should the chamber have so that Mr. Star can buy the least amount of barbed wire? Homework Help ✎

You are being asked to 'minimize' perimeter. Did you recognize that this is an optimization problem?
The first thing you will do is build a geometric equation that represents the situation. Then you will apply Calculus.

What do we know?

$\text{Area}: bh=40\rightarrow b=\frac{40}{h}$

$\text{Perimeter (there are only three sides)}: P=b+2h=\frac{40}{h}+2h$

Now that we have an equation, P(h), let's find it's minimum value.

$P(h)=\frac{40}{h}+2h$

$P^\prime(h)$ = ___________________

To find the h-value of the max and min, set $P^\prime(h)=0$, and solve for h.
Then, for a complete answer, don't forget to find the corresponding value of b.