This was the example that teacher gave us. A rectangular storage container with an open top is to have a volume of 10 m3. The length of this base is twice the width. Material for the base costs $\$15$ per square meter. Material for the sides costs $\$10$ per square meter. Find the cost of materials for the cheapest such container.
I solved finding first constrain equation which is $v=lwh= 5=w^2 h$.
And the the objective minimisation cost, which is $\text{cost}=(\text{area})(\text{price per }m^2)$.
Which is $36wh + 20w^2$, combing both equations I found $c= \frac{180}{w} +20w^2$.
Now he asked us and says the pyramid on the top of that rectangular container and express the cost as a function of the width of the container's shortest side.
The pyramid also has same height as the container.
The cost of the each sides of the pyramid is $18$ per meter square.
Total volume and all other informations are the same with the class example.
Please help me how do I found total volume in these two different situations. I’m really so confused. Thank you