Question

A rectangular box with a square bottom and no top has a volume of 2048 cubic inches. What values of the length x of a side of the bottom and the height y give the box with the smallest total surface area (the area of the bottom plus the area of the sides)

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Answer 1

The area of the rectangular box can be calculated using the given equation A = 4xy + $x^{2}$.

Explanation:

• Here 4xy represents area of the sides while $x^{2}$ represents the total area of the bottom.

• While the volume is 2048 = $x^{2} y$, if we solve y this gives us $y = 2048/x^{2}$

• Putting the value of y in the first equation, we get A =$4 x \cdot \dfrac{2048}{x^2} + x^2 = \dfrac{8192}{x} + x^2.$

• If x greater than zero, using the second derivative test we get $A' = -\dfrac{8192}{x^2} + 2 x, A'' = \dfrac{16384}{x^3} + 2.$

• Now finding the values of x and y, we get $\eqalign{ -\dfrac{8192}{x^2} + 2 x & = 0 \cr \noalign{\vskip2pt} 2 x & = \dfrac{8192}{x^2} \cr 2 x^3 & = 8192 \cr \noalign{\vskip2pt} x^3 & = 4096 \cr x & = 16 \cr} \\ y = \dfrac{2048}{256} = 8.$

• Applying the value of x we get  $A''(16) = \dfrac{16384}{4096} + 2 = 6 > 0.$