For a rectangular cube with a square base, suppose that it costs $3/cm^2 for the material used on the side and $6/cm^2 for the material used for the top lid and the base. assuming that the volume of this container is 54 cm^3 , what is the side length of the cube with the smallest cost?

The sides of the container should be 3 cm and height should be 6 cm to minimize the cost

Step-by-step explanation:

Data provided in the question:

costs  for the material used on the side = $3/cm²

costs for the material used for the top lid and the base = $6/cm²

Volume of the container = 54 cm³

Now,

let the side of the base be 'x' and the height of the box be 'y'

Thus,

x × x × y = 54 cm³

or

x²y = 54

or

y =  ............(1)

Now,

The total cost of material, C

C = $3 × ( 4 side area of the box ) + $6 × (Area of the top and bottom)

or

C = ( $3 × 4xy ) + ( $6 × (x²)  )

substituting the value of y in the above equation, we get

C =

or

C =

Differentiating with respect to x and putting it equals to zero to find the point of maxima of minima

thus,

C' = = 0

or

or

24x³ = 648

or

x = 3 cm

also,

C'' =

or

C''(3) = > 0

Hence,

x = 3 cm is point of minima

Therefore,

y =               [from 1]

or

y =

or

y = 6 cm

Hence,

The sides of the container should be 3 cm and height should be 6 cm to minimize the cost

d

step-by-step explanation:

hello and welcome!

so, when we see the words "proportional" along with a table, we need to know that we're looking numbers that divide by the same thing.

looking at this table, we simply have to do a trial and error, and divide to see which numbers have the same quotient. if you look at choice a & c, they both have equal quotients for three of the numbers, with one exception, so those can't be the answer. looking at b, there is no similarity, so that can't be it. then, we see d; if you divide all the numbers in the table, you'll see that they all divide by two, which is a proportional relationship, and our answer!

also, always remember that the x will always be the number that goes into y, and somewhere behind this table, is a formula. for this specific table, the formula would be:

since our x-value is being divided by two to get the y-value. and if they were to ask for more numbers in the series, this is the formula you would use to find them.