Multiple choice

An open rectangular tank with a square base and $32c.c$. of capacity has least surface area in $sq\ . cms$. is

  1. $48$
  2. $16$
  3. $32$
  4. $12$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let base side be x and height be h. Volume V = x^2 * h = 32, so h = 32/x^2. Surface area S = x^2 + 4xh = x^2 + 4x(32/x^2) = x^2 + 128/x. To minimize, dS/dx = 2x - 128/x^2 = 0. 2x^3 = 128, x^3 = 64, x = 4. Min area = 4^2 + 128/4 = 16 + 32 = 48.

AI explanation

For a square base of side s and height h, the volume is s squared times h equals 32. Using the surface area formula, s squared plus 4 times s times h, and substituting h as 32 divided by s squared, the derivative of area with respect to s is zero when s equals 4. The height is 32 divided by 16, which is 2 cm, making the least surface area 16 plus 32, which equals 48 sq. cm.

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