A rectangular tank with a square base, an open top, and a volume of 19 comma 65219,652 ft cubedft3 is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area.

34 and 17 feet.

Step-by-step explanation:

We have that the volume is given by:

V = x * x * y = (x ^ 2) * y

The surface area of the tank would be: x ^ 2 + x * y + x * y + x * y + x * y

SA = x ^ 2 + 4 * x * y

we know that the volume is 19652, replacing it would remain:

19652 = (x ^ 2) * y, we solve for y:

y = 19652 / (x ^ 2)

replacing in SA we are left with:

SA = x ^ 2 + 4 * x * 19652 / (x ^ 2)

SA = x ^ 2 + 70608 / x

we derive and equal 0:

SA' (x) = 2 * x - 70608 / (x ^ 2) = 0

2 * x = 70608 / (x ^ 2)

x ^ 3 = 39304

x = 34

for and would be:

y = 19652 / (34 ^ 2) = 17

dimensions are 34 and 17 feet.