Question

What is the rate of change of volume of a cube with respect to an edge when the diagonal of the cube is $6\sqrt{3}$ ?

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Answer 1

We need to find the rate of change of volume with respect to one edge. For this, we are given the diagonal of the cube. Using this, we can find the edge/side of the cube.

If a is the side of the cube then diagonal $a\sqrt{3}$

This is given as $6\sqrt{3}$

$\Rightarrow$ a = 6

Volume of a cube is $a^3$

Rate of change or instantaneous rate of change $li{m}_{h\to 0}\frac{f(a+h)-f\left(a\right)}{h}$

In this case the function f(x) is the volume of cube.

$\Rightarrow f\left(a\right)=a^3$

$\Rightarrow \text{rate of change}=li{m}_{h\to 0}\frac{f(a+h{)}^3-a^3}{h}=3a^2=3\times 6^2=108$

( You can also take the derivative of f(x) to get rate of change. That is the derivative of $a^3$, which will give you 3$a^2$)