Question

A company plans to manufacture a rectangular box with a square base, an open top, and a volume of 452 cm3. The cost of the material for the base is 0.4 cents per square centimeter, and the cost of the material for the sides is 0.6 cents per square centimeter. Determine the dimensions of the box that will minimize the cost of manufacturing it. What is the minimum cost

Answer 1

Answer:

The box has sides of 11.07 cm and height of 3.69 cm.

The cost (minimum) is 147 cents per box.

Step-by-step explanation:

We have a box with open top, with a volume of 452 cm^3.

Let x: base side of the box, in cm, and y: height of the box, in cm.

Then, the volume can be expressed as:

$V=x^2\cdot y=452\\\\y=452x^{-2}$

This box has 4 sides and 1 base. The material cost is 0.4 cents/cm^2 for the base and 0.6 cents/cm^2 for the sides.

Then, we can write the cost as:

$C=0.4\cdot 1\cdot (x^2)+0.6\cdot 4\cdot (xy)\\\\\\xy=x\cdot(452x^{-2})=452x^{-1}\\\\\\C=0.4x^2+2.4(452x^{-1})\\\\\\C=0.4x^2+1084.8x^{-1}$

The value for x that gives a minimum cost can be found deriving the function C and equal to 0:

$\dfrac{dC}{dx}=0.4(2x)+1084.8(-1\cdot x^{-2})=0\\\\\\0.8x-1084.8x^{-2}=0\\\\0.8x=1084.8x^{-2}\\\\0.8x^{1+2}=1084.8\\\\x^3=1084.8/0.8=1356\\\\x=\sqrt[3]{1356}\\\\x=11.07$

The height can be calculated with the equation:

$y=452x^{-2}=452(11.07^{-2})=452\cdot 0.00816 =3.69$

The minimum cost can be calculated as:

$C=0.4x^2+1084.8x^{-1}\\\\C(11.07)=0.4(11.07)^2+1084.8(11.07)^{-1}\\\\C(11.07)=0.4\cdot 122.51+1084.8\cdot0.09\\\\C(11.07)=49+98\\\\C(11.07)=147$