a rectangular tank with a square base, an open top, and a volume of 2048 ft³ is to be constructed of sheet…

Explanation:

Step1: Set up the volume and surface area formulas

Let the side length of the square base be (x) (in feet) and the height of the tank be (y) (in feet). The volume (V=x^{2}y), and since (V = 2048), we have (y=\frac{2048}{x^{2}}). The surface area (S=x^{2}+4xy) (because the base is (x\times x) and there are 4 sides each of area (x\times y)).

Step2: Substitute (y) into the surface - area formula

Substitute (y = \frac{2048}{x^{2}}) into (S): (S=x^{2}+4x\times\frac{2048}{x^{2}}=x^{2}+\frac{8192}{x}), where (x>0).

Step3: Find the derivative of (S) with respect to (x)

Using the power rule, if (S(x)=x^{2}+8192x^{- 1}), then (S^\prime(x)=2x - 8192x^{-2}=\frac{2x^{3}-8192}{x^{2}}).

Step4: Set the derivative equal to zero and solve for (x)

Set (S^\prime(x) = 0), so (\frac{2x^{3}-8192}{x^{2}}=0). Since (x^{2}>0) for (x>0), we solve (2x^{3}-8192 = 0). (2x^{3}=8192), then (x^{3}=4096), and (x = 16).

Step5: Find the value of (y)

Substitute (x = 16) into (y=\frac{2048}{x^{2}}). (y=\frac{2048}{16^{2}}=\frac{2048}{256}=8).

Answer:

(16,8)