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abc =32 ft³ is the maximum volume of an open rectangular box (with no top face) if its surface area is 1 square foot.
Calculate a square's area?
A rectangle with all equal sides, commonly known as a square. Multiplying the length by the length is the. Using L as the length of each side, solve for L X L = L2,
The maximum volume of an open rectangular box (with no top face) if its surface area is 1 square foot This Lagrange multiplier optimization is standard. If the box has a base of a, a height of c, and an area constraint of ab+2ac+2bc−48=0 we wish to optimize V= abc.
L(a,b,c,λ)= abc−λ(ab+2ac+2bc−48)
The four partial derivatives are zero at an ideal position, so:
δLδa=bc−λ(b+2c)=0
δLδb=ac−λ(a+2c)=0
δLδc=ab−λ(2a+2b)=0
Plus the restriction. The first two enlighten:
λ=bcb+2c=aca+2c
Consequently, b(a+2c)=a(b+2c) implies to b=a. The third partial, where b=a, now informs us that a2=4aλ and so λ=a/4 nd by using this information in the second partial, we obtain 4c=a+2c which informs us that c=a/2 .
Now that we've inserted these b and c expressions into the constraint, we get [tex]3a^2=1[/tex] which means that a=4 feet, b=4 feet, and c=2 feet.
The maximum volume is therefore, abc=32 ft³
Learn more about surface area here:
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