a rectangular tank that is 256 ft³ with a square base and open top is to be constructed of sheet steel of a…

Explanation:

Step1: Define variables and equations

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 = 256), we have (y=\frac{256}{x^{2}}). The surface - area (S) (amount of material) of the open - top tank is (S=x^{2}+4xy) (the base area (x^{2}) and four lateral faces each of area (xy)).

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

Substitute (y=\frac{256}{x^{2}}) into (S): (S(x)=x^{2}+4x\cdot\frac{256}{x^{2}}=x^{2}+\frac{1024}{x}), where (x>0).

Step3: Find the derivative of (S(x))

Using the power rule, if (S(x)=x^{2}+1024x^{-1}), then (S^\prime(x)=2x - 1024x^{-2}=\frac{2x^{3}-1024}{x^{2}}).

Step4: Find the critical points

Set (S^\prime(x) = 0), so (\frac{2x^{3}-1024}{x^{2}}=0). Since (x^{2}>0) for (x>0), we solve (2x^{3}-1024 = 0). (2x^{3}=1024), then (x^{3}=512), and (x = 8).

Step5: Use the second - derivative test

Find the second - derivative (S^{\prime\prime}(x)=2 + 2048x^{-3}). When (x = 8), (S^{\prime\prime}(8)=2+\frac{2048}{512}=2 + 4=6>0). So (S(x)) has a minimum at (x = 8).

Step6: Find the value of (y)

Substitute (x = 8) into (y=\frac{256}{x^{2}}), we get (y=\frac{256}{64}=4).

Answer:

(8,4)