How do you find generalized equations when optimizing using calculus?

Question

We have a homework question that we're struggling with and can't seem to find answers to.

The problem involves minimizing lengths of strapping used to fasten different boxes down.

The box is always 2m wide, x meters long and y meters tall.

There's two straps along the width of the box and one strap along the length such that the total strapping used is equal to 2x+6y+2 meters.

For each 1.5 meter increment of length, another strap is added across the width: eg. when n=2, x is between 1.5 and 3 meters and the total strapping is equal to 2x+8y+4 and when n=3, x is between 3 and 4.5 meters and the total strapping is equal to 2x+10y+6.

The question I need help with asks to use calculus and optimization to find a relationship between V and x and to then find the equations that link L (the length of strapping) and x.

Following this, it asks to find a generalized equation for L in terms of V, x, and n to confirm the generalized equation for L in terms of x & n. This is also done "using calculus and optimization."

Any help would be appreciated.

Answer 1

Optimization is typically a max or a min. Take the derivative and set it equal to zero.