Question

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, find $\frac{dv}{dx}$ (in $mm^{3}/mm$) when $x = 5$ mm. v(5)= mm³/mm. explain the meaning of v(5) in the context of this problem. v(5) represents the rate at which the volume is increasing as x reaches 15 mm. v(5) represents the volume as the side length reaches 5 mm. v(5) represents the rate at which the volume is increasing with respect to the side length as v reaches 15 mm³. v(5) represents the rate at which the volume is increasing with respect to the side length as x reaches 5 mm. v(5) represents the rate at which the side length is increasing with respect to the volume as x reaches 5 mm.

Explanation:

Step1: Recall volume formula for cube

The volume $V$ of a cube with side - length $x$ is given by $V(x)=x^{3}$.

Step2: Differentiate the volume function

Using the power - rule for differentiation $\frac{d}{dx}(x^{n})=nx^{n - 1}$, if $V(x)=x^{3}$, then $\frac{dV}{dx}=V^{\prime}(x)=3x^{2}$.

Step3: Evaluate the derivative at $x = 5$

Substitute $x = 5$ into $V^{\prime}(x)$. So $V^{\prime}(5)=3\times5^{2}=3\times25 = 75$ $mm^{3}/mm$.

Answer:

$V^{\prime}(5)=75$ $mm^{3}/mm$
The meaning of $V^{\prime}(5)$: $V^{\prime}(5)$ represents the rate at which the volume is increasing with respect to the side length as $x$ reaches $5$ mm. So the correct option is: $V^{\prime}(5)$ represents the rate at which the volume is increasing with respect to the side length as $x$ reaches $5$ mm.