Using differentials to approximate increments to find the amount of material in a hollow rectangular box, whose inside measurements are 5ft long, 3ft wide, and 2ft deep, if the box made of lumber that is 0.5inche thick and the box has no top?
what I understand in this question is $x=5$ , $y=3 $, $z=2$ and the function that must be used to find the volume is this$f(x,y,z)=x*y*z$ and the thickness is $0.5 inches$ so we have to find the volume with respect to that as well
I want the answer with steps and by using partial derivatives and I don't even know where to start and what to do!
use this:- $$df=\frac{\partial{f}}{\partial{x}}dx+\frac{\partial{f}}{\partial{y}}dy+\frac{\partial{f}}{\partial{z}}dz $$
$$0.5in=\frac{1}{24}ft$$
In my textbook, it requires the change of volume (delta V) and the answer is ($\frac{47}{24}$) but I don't know the solution.
Let us denote the thickness by $a$. The volume of the material is equal to \begin{align} \Delta V = (x+2a)(y+2a)(z+a) -xyz &\approx a\cdot \frac{d}{dt}\Big|_{t=0}\big((x+2t)(y+2t)(z+t)\big) = \\ &= a \cdot\Big(2\frac{\partial (xyz)}{\partial x} + 2\frac{\partial (xyz)}{\partial y} + \frac{\partial (xyz)}{\partial z}\Big) = \\ &= a(2yz+2xz+xy) = \\ &= \frac{47}{24} \text{ ft}^3\end{align} The result doesn't quite match the answer you're given (47 vs 27), but I believe this is a typo in your book.