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

step-by-step explanation:

answer: c, the green line

step-by-step explanation:

there is no black line, eliminate b.

simple interest implies periodic payments, but the red line shows no payments, eliminate d.

simple interest has equal payments. if paid say every day, the graph is about a straight line (actually a stairstep).

the yellow line is not straight, eliminate a.

check: the green line is straight, answer is c.

simple interest with positive interest rate gives a straight line sloping up to the right. green line:

g(t) = $100 + ($200-$100) ÷ (20 years) × (t years)

= $100 + (4%/year) × ($100)(t years)

incidentally, the yellow line curving upward represents continuously compounded interest

y = $100 × 2^(t years / (14 assuming the yellow line hits $200 at 14 years. 2^0 is 1, so y(0) = $100; 2^(14/14) is 2, so y(14) = $200.