Question

An open top rectangular box with a square bottom has a volume of 120 cubic meters. Its bottom and sides are made from two different materials. It costs 10 dollars per square meter for the bottom material and 12 dollars per square meter for the sides.

Answer 1

Answer:

The dimensions that minimizes the cost are a= 6.604 (lenght of a side) and h=2.752 (height)

Step-by-step explanation:

Let's define some notation first:

h = represent the height of the box

a= length of a side on the bottom

We need to remember that we have on this case an open box, so then we don't have a top part.

Let's begin with the formula for the volume of the box, we have a total volume given of $120m^3$, and we know that the volume for a box is the product of the area and the height like this:

$120 m^3 = a^2 h$   (1)

From equation (1) we cna solve for the height and we got:

$h=\frac{120}{a^2}$   (2)

Now let's move to th surface area formula. The top bottom area is $a^2$ and the surface area for the four lateral sides is 4ah. Since we know that the cost for the bottom material is 10 $ per each square meter and for the sides is 12$ per square meter we can create a function cost like this:

$Cost= Cost of bottom+ Cost of sides$

$Cost = 10a^2 + (12)(4ah) = 10a^2 + 48ah$   (3)

Now we can substitute into equation (3) the equation (2) like this:

$Cost = 10a^2 + (12)(4ah) = 10a^2 + 48a\frac{120}{a^2}=10a^2 +\frac{5760}{a}$   (4)

Now in order to find the minimum cost, we can take the first derivate for the expression (4) like this

$\frac{d}{da} Cost = 20a -\frac{5760}{a^2}$

We can set this equal to 0 and solve for a and we got:

$\frac{5760}{a^2}= 20a$

$\frac{5760}{20}=a^3$

$a=6.604$

And since we have a we can find h using the expression (2) like this:

$h=\frac{120}{a^2}=\frac{120}{6.604^2}=2.752$  

So then the dimensions that minimizes the cost are a= 6.604 (lenght of a side) and h=2.752 (height)