An open-topped glass aquarium with a square base is designed to hold $4$ cubic feet of water. What is the minimum exterior surface area of the aquarium? $\newline$ square feet

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

4

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

\newline

square feet

Define Volume Formula: Let $s$ be the side length of the square base of the aquarium. $\newline$ The volume of the aquarium is the cube of the side length since it's a square base. $\newline$ $V = s^3$
Substitute Volume Value: Volume of aquarium: $4$ cubic feet $\newline$ Substitute $4$ for $V$ in $V = s^3$ . $\newline$ $4 = s^3$
Find Side Length: Find the side length $s$ . $\newline$ $4 = s^3$ $\newline$ $s = \text{cube root of } 4$ $\newline$ $s \approx 1.5874$ feet (rounded to $4$ decimal places)
Calculate Surface Area Formula: Calculate the exterior surface area of the aquarium. $\newline$ The surface area of a square-based open-topped box is the area of the base plus the area of the four sides. $\newline$ Surface Area = $s^2 + 4(s \times h)$ $\newline$ But since the height $h$ is the same as the side length $s$ (because the volume is a cube of the side length), we can simplify this to: $\newline$ Surface Area = $s^2 + 4(s \times s)$ $\newline$ Surface Area = $s^2 + 4s^2$ $\newline$ Surface Area = $5s^2$
Substitute Side Length: Substitute the side length $s$ into the surface area formula. $\newline$ Surface Area $= 5 \times (1.5874)^2$ $\newline$ Surface Area $\approx 5 \times 2.5197$ $\newline$ Surface Area $\approx 12.5985$ square feet