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A rectangular storage container with an open top is to have a volume of 10m3.?
A rectangular storage container with an open top is to have a volume of 10m3. The length of it's base is twice the width. Material for the base costs $ 10 per square meter. Material for the sides costs $ 6 per square meter. Find the dimentions and the cost of materials for the cheapest such container.
Width = ?
Length =?
Height =?
Cost $ =?
Let w = width (in meters) of the box; then the length is (2w) meters.
Let h = height (in meters) of the box. Since the volume is to be 10 m³, and the formula for the volume V of a rectangular box is
V = length×width×height
we have
10 = (2w)(w)(h)
Solving for h,
h = 5/w²
The area of the bottom is (length×width) = 2w²
The cost (in dollars) of the material for the bottom is 10*2w² = 20w²
The area of each of two sides of the box is (length×height) = (2w)(5/w²) = 10/w
The area of each of the other two sides of the box is (width×height) = w(5/w²) = 5/w
The total area of the four sides is 2(10/w + 5/w) = 30/w
The cost (in dollars) of the materials for the four sides is 6(30/w) = 180/w
The total cost C of materials is thus given by
C = 20w + 180/w
dC/dw = 20 - 180/w²
dC/dw = 0 ⇒ 20 - 180/w² = 0
⇒ w² = 9
⇒ w = ± 3
A negative value for w makes no physical sense, so we discard this solution. Thus,
w = 3 meters.
You should confirm using the second derivative test that this is a minimum. Then you can finish the problem.
AGRO-INDUS Square Silo System (Grain Storage Solution).mp4