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An open-topped glass aquarium with a square base is designed to hold 32 cubic feet of water. What is the minimum possible exterior surface area of the aquarium?
square feet
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An open-topped glass aquarium with a square base is designed to hold $32$ cubic feet of water. What is the minimum possible exterior surface area of the aquarium? $\newline$ square feet $\newline$ Stuck? Review related articles/videos or use a hint. $\newline$ Report a problem

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Understanding integers

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Q. An open-topped glass aquarium with a square base is designed to hold

32

cubic feet of water. What is the minimum possible exterior surface area of the aquarium?

\newline

square feet

\newline

Stuck? Review related articles/videos or use a hint.

\newline

Report a problem

Volume Equation: Let's denote the side of the square base as $s$ feet. Since the aquarium has a square base and is open-topped, its volume $V$ is given by the formula $V = s^2 \cdot h$ , where $h$ is the height of the aquarium in feet. $\newline$ We know the volume of the aquarium is $32$ cubic feet, so we can write the equation: $\newline$ $s^2 \cdot h = 32$
Minimizing Surface Area: To minimize the surface area, the aquarium should be shaped as a cube, because a cube has the smallest surface area for a given volume. This means that the height $h$ should be equal to the side length $s$ . So, we can set $h = s$ and rewrite the equation as: $s^2 \cdot s = 32$ $s^3 = 32$
Calculating Side Length: Now we solve for $s$ by taking the cube root of both sides of the equation: $\newline$ $s = 32^{1/3}$ $\newline$ Calculating the cube root of $32$ gives us: $\newline$ $s \approx 3.1748$ feet (rounded to four decimal places for precision in further calculations)
Calculating Surface Area: Now that we have the side length $s$ , we can calculate the exterior surface area of the aquarium. The surface area $A$ of an open-topped cube with side length $s$ is given by the formula: $\newline$ $A = 4 \cdot s^2 + s^2$ (the area of the four sides plus the base) $\newline$ Substituting the value of $s$ we found: $\newline$ $A = 4 \cdot (3.1748)^2 + (3.1748)^2$
Calculating Surface Area: Now that we have the side length $s$ , we can calculate the exterior surface area of the aquarium. The surface area $A$ of an open-topped cube with side length $s$ is given by the formula: $\newline$ $A = 4 \times s^2 + s^2$ (the area of the four sides plus the base) $\newline$ Substituting the value of $s$ we found: $\newline$ $A = 4 \times (3.1748)^2 + (3.1748)^2$ Performing the calculations: $\newline$ $A = 4 \times (10.0774) + 10.0774$ $\newline$ $A = 40.3096 + 10.0774$ $\newline$ $A = 50.387 \text{ feet}^2$

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