My question is
The volume of a cube is increased from $1000$ cubic centimeters to $1156$ cubic centimeters.
Use differentials to determine. the side length of the cube increases by? the surface area increases by? and the surface area of the cube increases by what $\%$?
For, "how many centimeters does the side length increase by?" I did:
$$v=x^3$$
$$\sqrt[3]{1000}=x$$
$$x=10cm$$
So $\sqrt[3]{1156}= 10.495cm$ now $ 10.495cm-10cm=0.495$ but the correct answer is $0.52cm$, what do I have to do differently.
For, "how many square cm does the SA increase?" I did:
$SA=6x^2$. plugged in $x$ for both the volumes and calculated the dif to be $60.88cm^2$ but the actual answer is $62.4cm^2$
and since I got that wrong the $\%$ increase was obviously wrong.
Any idea on what I should do differently?
Best Answer
For a cube with side length $x$ and volume $V$, we have $x=V^{\frac13}$.
So $dx=\frac13 V^{-\frac23} dV$.
You would use the values of $V=1000$ and $dV=156$ to find $dx$.