a rectangular tank that is 2916 ft³ with a square base and open top is to be constructed of sheet steel of a…

Explanation:

Step1: Define variables

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 = 2916), we have (y=\frac{2916}{x^{2}}). The surface area (S=x^{2}+4xy) (because the top is open).

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

Substitute (y=\frac{2916}{x^{2}}) into (S): (S=x^{2}+4x\cdot\frac{2916}{x^{2}}=x^{2}+\frac{11664}{x}), where (x>0).

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

Using the power rule, (S^\prime(x) = 2x-\frac{11664}{x^{2}}).

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

Set (S^\prime(x)=0): [ \begin{align*} 2x-\frac{11664}{x^{2}}&=0\ 2x&=\frac{11664}{x^{2}}\ 2x^{3}&=11664\ x^{3}& = 5832\ x&=18 \end{align*} ]

Step5: Find the second - derivative of (S) and check for minima

Find (S^{\prime\prime}(x)=2+\frac{23328}{x^{3}}). When (x = 18), (S^{\prime\prime}(18)=2+\frac{23328}{18^{3}}=2 + 4=6>0). So (S(x)) has a minimum at (x = 18).

Step6: Find the value of (y)

Substitute (x = 18) into (y=\frac{2916}{x^{2}}), (y=\frac{2916}{18^{2}}=\frac{2916}{324}=9).

Answer:

(18,9)