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What is the smallest parameter possible for a rectangle whose area is 16 square inches and what are it’s dimensions?

Answer 1

Scarlett Allison

We get #a=b=4#

We have the Perimeter as

#p=2(a+b)=2(a+16/a)# So we get by #AM-GM#

#a+16/a>=2sqrt(a*16/a)=8# Multiplying by #2#

#2(a+16/a)>=4sqrt(16)=16# the equal sign holds if #a=b=4#

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Answer 2

Emilia Aguilar

#P=16#

Let #a=x# inches be one side of the rectangle, then the other side is:

#b=16/x#

measured also in inches.

The perimeter is then:

#P= 2(x+16/x)#

Evaluate the derivative:

#(dP)/dx = 2-32/x^2#

thus the derivative is null for #x=4#, and in this point the second derivative:

#(d^2P)/dx^2 = 64/x^3 > 0#

thus #x=4# is a minimum.

We can conclude that the smallest perimeter is obtained when #a=b=4#, that is when the rectangle is a square.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.