A rectangular storage container with an open top is to have a volume of 10 m^3 . The length of its base is three times the width. Material for the base costs $10 per square meter.

Question

A rectangular storage container with an open top is to have a volume of 10 m^3. The length of its base is three times the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container.

Answer 1

(No Calculus Method)

We can use the equations

3w*w*h = 10

3w*w*10 = C_b

(2hw + 2h(3w))*6 = C_s

where w is the width, h is the height, C_b is the cost of the base and C_s is the cost of the sides to get a function that shows the total cost of the materials (C_b + C_s) in terms of one variable.

We can say h = 10/(3w²) from the first equation which means C_s = 160/w. The cost function we should look at in this case is C_t = 30w² + 160/w.

After graphing this function, we can see that there is one local minimum on the positive side at w = 1.4, C_t = 173.1. Thus, our answer for the cost of materials is $173.

Answer 2

3w*w*h=1000m

Need more imfo.