## Related questions with answers

Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate dV/dx when x = 3 mm and explain its meaning.

Solutions

Verified$V = x^3$ so $\dfrac{dV}{dx} = 3x^2$

We have, at $x= 3$ mm $(\dfrac{dV}{dx})_{x=3} = 3 \cdot 3^2 = 27$ square mm.

$dV/dx$ measures the change in volume for a unit change in the length, and this value is dependent on the instantaneous value of the length.

At $x=3$, this rate measures that the volume would increase by 27 mm cubed if the length changed by 1 mm, when the length was already 3 mm.

The volume of a cube with side length $x$ is

$V(x)=x^3$

Differentiate with respect to $x$

$\dfrac{dV}{dx}=\dfrac{d\left(x^3\right)}{dx}$

Use the power rule for differentiation

$\dfrac{dV}{dx}=3x^{3-1}=3x^2$

Substitute $x=3$

$\dfrac{dV}{dx}\bigg|_{x=3}=3\cdot3^2=3^3=27\text{ mm}^2$

This means that when the crystals reach a side length of 3mm, a 1 unit increase in the side length will result in a volume increase of 27 unit$^3$

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