Answer:
Base=3.42cm
Height = 10.26 cm
For a total cost of $631.59
Step-by-step explanation:
In order to solve this problem we can start by drawing what the box will look like. (See attached picture).
So, the problem mentions that the box will have a volume of [tex]120cm^{3}[/tex]. Since the box has a squared base, we can calculate its volume by using the following formula:
[tex]V=b^{2}h[/tex]
we can turn this into an equation by substituting the corresponding volume:
[tex]b^{2}h=120[/tex]
now, the problem tells us that the base of the box will cost [tex]$18/cm^{2}[/tex] so first, we need to calculate the area of the base of the box. Its area can be calculated by using the following formula:
[tex]A_{base}=b^{2}[/tex]
so the cost of the base is calculated with the following equation:
[tex]C_{base}=18b^{2}[/tex]
we can find the cost of the sides of the box by following a similar procedure:
[tex] A_{side}=bh[/tex]
since there are 4 sides to the box, then we can calculate the total area of the sides by multiplying the formula by 4.
[tex]A_{sides}=4bh[/tex]
the problem tells us that the cost of the sides of the container is: [tex]$3/cm^{2}[/tex] so the cost of the sides will be:
[tex]C_{sides}=3(4)bh[/tex]
[tex]C_{sides}=12bh[/tex]
So the total cost of the box is found by adding the two costs we just found:
[tex]C_{total}=C_{base}+C_{sides}[/tex]
[tex]C_{total}=18b^{2}+12bh[/tex]
so we can take the volume equation to find an equation we can substitute for h on the cost equation:
[tex]h=\frac{120}{b^{2}}[/tex]
when substituting we get:
[tex]C_{total}=18b^{2}+12b(\frac{120}{b^{2}})[/tex]
Which simplifies to:
[tex]C_{total}=18b^{2}+\frac{1440}{b}[/tex]
or:
[tex]C_{total}=18b^{2}+1440b^{-1}[/tex]
In order to minimize the costs we will now take the derivaative of this function and set it to be equal to zero:
[tex]C'=36b-1440b^{-2}[/tex]
[tex]36b-1440b^{-2}=0[/tex]
and now we solve for b:
[tex]36b=\frac{1440}{b^{2}}[/tex]
[tex]36b^{3}=1440[/tex]
[tex]b^{3}=\frac{1440}{36}[/tex]
[tex]b^{3}=40[/tex]
[tex]b=\sqrt[3]{40}[/tex]
b=3.42 cm
so now we can use this value to find the height:
[tex]h=\frac{120}{3.42}[/tex]
h=10.26 cm
the total cost is found by using the cost equation:
[tex]C_{total}=18b^{2}+1440b^{-1}[/tex]
[tex]C_{total}=18(3.42)^{2}+1440(3.42)^{-1}[/tex]
[tex]C_{total}=$631.59[/tex]
