How do I make a function smaller? I'm doing a work on calculating the volume necessary to build the carcass of the starship enterprise, my plan is to do it first using the surface area formula (which I already did) and then calculating the integral volume of the ship and subtracting from it a volume which is 5 centimeters smaller. As I already have all the functions to calculate it in its orginal size how can I make it 5cm smaller? (I also have the points which I used to make the regression and find the functions)
 A: A good approximation can be done very easily. Take a measure of the ship that is midly affacetd by those $5$ cm. Don't choose the longest nor the shortest dimension.
Now solve this proportion:
$$\frac{\text{Volume}}{\text{Inner volume}}=\left(\frac{\text{Chosen measure}}{\text{Chosen measure$-5$ cm}}\right)^3$$
A: It sounds like what you want is the volume of a shell that is the outer surface of the ship and $5$ cm thick.  As long as the important dimensions of the ship, you will have a good approximation by multiplying the surface area of the ship by $5$ cm.  The corresponding calculation on a sphere goes like this.  If the original radius is $R$ the shell is the space between that sphere and a sphere of radius $R-5$.  The volume is then $\frac 43\pi(R^3-(R-5)^3)=\frac 43\pi(15R^2-75R+125)$  For large $R$, the last two terms can be ignored, leaving $4\pi R^2\cdot 5$, which if $5$ times the surface area of the sphere.  The same goes on with any shape as long as the dimensions are large compared to the thickness of the shell.
