In a manufacturing process the boxes with nominal dimensions of $5$ inches by $5$ inches by $2$ inches are subject to an error of $1\%$ in each dimension.Use differentials to estimate the error in volume of the box. Compare it with the actual minimum and maximum value of the box.
I do not know how we can solve such a problem. I am completely lost. Please help me.
The volume of a box with dimensions $a$, $b$, $c$ is given by $$V(a,b,c)=a\>b\>c\ .$$ It follows that $${dV(a,b,c)\over V(a,b,c)}={da\over a}+{db\over b}+{dc\over c}\ .$$ Therefore, "in first approximation", the relative error in volume is the sum of the relative errors in the side lengths.
In the given example the relative errors in the side lengths are $1\%$. As the case may be these add up, and we obtain a relative volume error of $3\%$, or $1.5$ cubic inches as a "first estimate".
The actual minimal volume of the box under the given constraints is $$4.95\cdot 4.95\cdot1.98=48.51495$$ cubic inches, which is $<1.5$ cubic inches off; and the maximal volume is $$5.05\cdot5.05\cdot2.02=51.51505$$ cubic inches, which is $1.51505>1.5$ off the intended value of $50$ cubic inches. It follows that the worst case analysis gives a larger error than the "first estimate".