Estimating volume of cube by differentiation

Question

I'm looking for guidance as to how to go about this problem.

A cube with $10$ inch sides is coated with $0.2$ fiberglass. Use differentials to estimate the volume of the fiberglass shell.

I know that the formula for the volume of the cube is

$$V = x^3$$

and that by differentiating, it would be

$$\frac{\mathrm{d}V}{\mathrm{d}x} = 3x^2$$

Answer 1

That's correct way to start, then we have

$$V(10+\Delta x)=V (10)+V'(10)\cdot\Delta x\implies \Delta V=V'(10)\cdot\Delta x$$

Note also that we need to assume $\Delta x=2\cdot 0.2=0.4$.

Then we have

• $V'(10)=300$
• $\Delta x= 0.4$
• $\Delta V=120$

Note that we can also calculate by the area of the cube times the thickness of the coating and we obtain

• $\Delta V=A\cdot t=6\cdot 100 \cdot 0.2=600\cdot 0.2 = 120$

Answer 2

Assuming "0.2 fiberglass" is $0.2$ inch thick, we are interested in $V(10.4)-V(10)$ (since the cost is applied to both sides). Considering the definition of $V'(x)$, we have $$ V(10.4)-V(10)\approx V'(10)\cdot 0.4 $$

Answer 3

That's right. So now you want to estimate $(10.4)^3-10^3$ by using that derivative. That is, you want to use $f(x+h)-f(x)\approx hf'(x).$

Answer 4

Your $3x^2$ is the rate of change of the volume with respect to the side of cube.

The linear approximation to the actual change is the rate of change times the change.

The change in the side of cube is $0.4\text { inch.}$

Thus the total change is $$ 3(10^2)(0.4)=120 \text { cubic inches}.$$