How do you find the dimensions of a rectangle whose area is 100 square meters and whose perimeter is a minimum?

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Answer 1 · 2014-10-08T02:06:06

Let #x# and #y# be the base and the height of the rectangle, respectively.

Since the area is 100 #m^2#,

#xy=100 Rightarrow y=100/x#

The perimeter #P# can be expressed as

#P=2(x+y)=2(x+100/x)#

So, we want to minimize #P(x)# on #(0,infty)#.

By taking the derivative,

#P'(x)=2(1-100/x^2)=0 Rightarrowx=pm10#

#x=10# is the only critical value on #(0,infty)#

#y=100/10=10#

By testing some sample values,

#P'(1)<0 Rightarrow P(x)# is decreasing on #(0,10]#.

#P'(11)>0 Rightarrow P(x)# is increasing on #[10,infty)#

Therefore, #P(10)# is the minimum

I hope that this was helpful.

Hence, the dimensions are #10\times10#.