Answer to Question #94604 in Calculus for Ikin

Question

3. A rectangular (uncovered) container can fill up 3000 cm3 of liquid that fixed with a length of 20cm. Find the dimensions of the container that has the least surface area.

Answer 1

Let length of the container = x = 20, width of the container = y, and height of the container = z.

Volume of the container $V=x\cdot y\cdot z=3000,\ y\cdot z=3000/20=150$

The surface of the container $S=x\cdot y+2(x+y)z=20y+2x\cdot z+2\cdot y\cdot z=$

$20y+40\cdot150/y+300=20y+6000/y+300$

Using inequality of arithmetic and geometric means

$20y+6000/y\ge 2\sqrt{20y\cdot 6000/y}=400\sqrt{3}$ , and equality possible when $20y=6000/y, y^2=300, y=\sqrt{300}$ , therefor the area of the container is minimal when

$y=\sqrt{300}=10\sqrt{3},\ z=150/(10\sqrt{3})=5\sqrt{3}$ .

Answer: dimensions of the container that has the least surface area are length = 20 cm, width = $10\sqrt{3}$ cm, height = $5\sqrt{3}$ cm.