The company Humpty Dumpty stock cylinder shaped metal boxes shall have a volume
ID: 3187926 • Letter: T
Question
The company Humpty Dumpty stock cylinder shaped metal boxes shall have a volume of 330ml (= 330 cm3), see figure below. The production method allows the circular bottom and top eyelids are twice as thick as the side. Since is 0.1mm thick, while the top and bottom is 0.2mm thick.
a) Let x be the radius in centimeters. Show that the total volume (in cm ^ 3) of metal needed to construct such cans as specified above is approximately given by:
b) Find the radius with the least consumption of metal per box.
Explanation / Answer
a) We know that the volume of the box is pi * x^2 * h = 330. This means that h = 330 / (pi * x ^ 2) The total metal M can be written as: M = 0.02 * (2 * area of circle) + 0.01 * (side surface area). The area of the circle is pi * x ^ 2. The side surface area is 2 * pi * x * h. Using the volume equation from above we get: SA = 2 * pi * x * (330 / (pi * x ^ 2)) = 2 * 330 / x. Thus, M = 0.02 * 2 * pi * x^2 + 0.01 * (2 * 330 / x), which is written above. b) Finding the least consumption of metal per box means that we need to minimize M. We do that by taking the first derivative of M and setting it equal to 0. M' = 0.02 * 4 * pi * x + 0.01 * 2 * 330 * (-x ^ -2) = 0 0.08 * pi * x - 6.60 x ^ -2 = 0. Multiply by x^2 on both sides: 0.08 * pi * x ^ 3 - 6.60 = 0 Solve for x: x = cube root(6.60 / 0.08 pi) ~= 2.97 Thus, the radius with with the least consumption of metal per box is approximately 2.97 cm.