Aclosed box with a square base is to have a volume of 85. 75 cm^3. the material for the top and bottom of the box costs $3.00 per square centimeter, while the material for the sides costs $1.50 per square centimeter. find the dimensions of the box that will lead to the minimum total cost. what is the minimum total cost?

We can create two equations here:

(1) Volume = area of square * height of box

85.75 = s^2 h

(2) Cost = 3 * area of square + 1.5 * area of side box

C = 3 s^2 + 1.5 s h

From (1), we get:

h = 85.75 / s^2

Combining this with (2):

C = 3 s^2 + 1.5 s (85.75 / s^2)

C = 3 s^2 + 128.625 s-

Taking the 1st derivative and equating dC/ds = 0:

dC/ds = 6s – 128.625 / s^2 = 0

Multiply all by s^2:

6s^3 – 128.625 = 0

6s^3 = 128.625

s = 2.78 cm

So h is:

h = 85.75 / s^2 = 85.75 / (2.78)^2

h = 11.10 cm

So the dimensions are 2.78 cm x 2.78 cm x 11.10 cm

The total cost now is:

C = 3 (2.78)^2 + 1.5 (2.78) (11.10)

C = $69.47

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step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for