A cone with a circular base has a height of 40 cm and its radius at the base is 15 cm. Each measurement has 0.3 cm precision. With the help of differentials, estimate the greatest error that is incurred on the volume of the cone.
I first started by determining what the function for the volume of a cone was. This was $πr^2⋅\frac{h}{3}$.
Now, for finding our differentials, we have that $f_r(r,h)=π2r\frac{h}{3}$ and for $f_h(r,h)=πr^2⋅\frac{1}{3}$.
Now, for our linerization function, we have $L(r,h)=f(a,b)+f_r(a,b)(r-a)+f_h(a,b)(h-a)$ which, having inputed all the variables gives us $L(r,h)=π225 \frac{40}{3} + π30 \frac{40}{3}(r-15) + \pi 225\cdot\frac{1}{3}(h-40)$.
My thought process was that, in finding the linearization for the function of the volume of the cone I could then simply find the difference between $L(15.3,40.3)$ and $L(15,40)$ which would give me some "error" volume.
Why/where is my this logic incorrect?
edit: See comments below. Note that reasoning is sound, but errors in differentiation as well as in the understanding of "linearization" ($x^2$ is not linear) led to incorrect answer.