Cost function for Minimizing cost of a hole 
I'm solving a problem where I have to minimize the cost of a hole. The depth $d$ needs to be 4 times the width $b$ of the hole. What should my cost function be?
This i what I have come up with:
Cost = $4d*b + 4d*b + 4d*b + 4d*b + b*b = 16db+b^2$
Volume = $b*b*d = b^2d$
Am i on the right track?
Thanks
 A: Your function for volume is correct. This is something you can use to compute the values for the length or depth with respect to the other variable since you already know the value for V.
Also, you know that $d = 4b$ per the last constraint. This reduces the problem down to a single variable optimization problem.
You function for cost is not correct; You want to minimize area while keeping volume constant.
A: If $d=4b$ and $b^2d=100000$, then we can solve a set of simultaneous equations to find our solution. $b^2(4b)=100000 \rightarrow 4b^3=100000 \rightarrow b^3=25000 \rightarrow b\approx 29.2401773821m$. This means that $d=4(29.2401773821)\approx 116.960709529m$. There are no other solutions, so this is the optimal (and only) choice to make.
If, however, $d\neq 4b$, we can still solve.
First, relate $b$ and $d$ in terms of each other. As $100,000=b^2d$, therefore, $d=\dfrac{100,000}{b^2}$. Plugging this in to the first equation gives us $C=16(\dfrac{100,000}{b^2})b+b^2 \rightarrow C=b^2+\dfrac{1,600,000}{b}$
To solve for the minimum cost, we can find the vertex of the created function, which occurs where $b \approx 92.8$. This means that the cheapest cost for the hole would be a hole with dimensions $92.8m*92.8m*11.6m$
