Find the cost of materials for the cheapest such container.

Question

A rectangular storage container with an open top is to have a volume of 10 𝑚³. The length of its base is twice the width. Material for the base cost $10 per square meter. Material for the sides cost $6 per square meter.

Answer 1

w*2w*y=10cu/m

2w^2*y=10

Need more imfo.

Answer 2

Let's call the overall cost C, the area of the base A_b, and the area of the sides A_s. Let's also call the length, width, and height l, w, and h. Then:

A_b = lw

A_s = 2hl + 2hw

C = cost of base + cost of sides = 10A_b + 6A_s = 10lw + 6(2hl + 2hw) = 10lw + 12hl + 12hw.

According to the problem, l = 2w, so we have:

C = 10(2w)w + 12h(2w) + 12hw = 20w² + 24hw + 12hw = 20w² + 36hw.

Because the volume is 10, we have:

10 = hlw = h(2w)w = 2hw²

5/w² = h

Substituting this equation into the previous result, we get:

C = 20w² + 36(5/w²)w = 20w² + 180/w.

Now we have C simply as a function of one variable (w). We need to find the lowest cost, so take the first derivative and set it equal to 0 to find the value of w that produces the minimum value of C. I think that you can take it from here. (The answer that I get is w = the cube root of 4.5, which can then be substituted into the previous equation to get C.)