4. A rectangular box with length X, width Y and height Z is to be builtwith a vo

Question

4. A rectangular box with length X, width Y and height Z is to be builtwith a volume of 10ft^3. The material to build the bottom costs $3/ft^2, and the material to build the four vertical sides and the top costs $2/ft^2. Find the length x, width Y and height Z of the least expensivebox that can be built with these materials (and use correct units in stating your values)

A rectangular box with length X, width Y and height Z is to be built with a volume of 10ft^3. The material to build the bottom costs $3/ft^2, and the material to build the four vertical sides and the top costs $2/ft^2. Find the length x, width Y and height Z of the least expensive box that can be built with these materials (and use correct units in stating your values)

Explanation / Answer

since xyz = 10

total cost c = 3xy +2x2(yz + xz) + 2xy

= 5xy + 4yz + 4xz

= 5 ( 10 / z) + 4 (10/x) + 4 (10 / y)

= 50 / z + 40 / x + 40 / y

since we have to manimize cost c

so dc / dz = 0 means 0 = 0 + 4y + 4z

or y = -z

also dc/ dy = 0 means 0 = 5x + 4z + 0

or x = -4z / 5

putting in equation first

(-4z/5)(-z)(z) = 10

or z^3 = 50/4

so z = (25/2)^1/3

and x = (4/5)(25/2)^1/3

and y = (25/2)^1/3

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