a rectangular tank with a square base, an open top, and a volume of 32 ft³ is to be constructed of sheet…

Explanation:

Step1: Set up the volume and surface area equations

Let the side length of the square base be (x) (in feet) and the height be (h) (in feet). The volume (V=x^{2}h), and since (V = 32), we have (h=\frac{32}{x^{2}}). The surface area (S=x^{2}+4xh) (because the tank has an open - top). Substitute (h=\frac{32}{x^{2}}) into the surface area formula: (S=x^{2}+4x\cdot\frac{32}{x^{2}}=x^{2}+\frac{128}{x}), (x>0).

Step2: Find the derivative of the surface area function

Differentiate (S(x)) with respect to (x). Using the power rule, if (y = x^{n}), then (y^\prime=nx^{n - 1}). (S^\prime(x)=2x-\frac{128}{x^{2}}).

Step3: Find the critical points

Set (S^\prime(x) = 0). [ \begin{align*} 2x-\frac{128}{x^{2}}&=0\ 2x&=\frac{128}{x^{2}}\ 2x^{3}&=128\ x^{3}&=64\ x& = 4 \end{align*} ]

Step4: Check the second - derivative for a minimum

Differentiate (S^\prime(x)) to get (S^{\prime\prime}(x)=2+\frac{256}{x^{3}}). When (x = 4), (S^{\prime\prime}(4)=2+\frac{256}{64}=2 + 4=6>0). So (S(x)) has a minimum at (x = 4).

Step5: Find the height

Substitute (x = 4) into (h=\frac{32}{x^{2}}). Then (h=\frac{32}{16}=2).

Answer:

(4,2)