Q 12. - Calculus

Question

Suppose the side lengths $x,y,$ and $z$ of a rectangular box are each functions of time.

$\left(a\right)$ How is the rate of change of the volume of the box related to the rates of change of $x,y,$ and $z$?

$\left(b\right)$ How is the rate of change of the surface area of the box related to the rates of change of $x,y$ and $z$?

Step-by-Step Solution

Answer

The required relationship between the rate of change of the volume of the rectangular box and the rate of change of $x,y$ and $z$ of the rectangular box is $V^{\text{'}}\left(t\right)=x^{\text{'}}\left(t\right)y\left(t\right)z\left(t\right)+x\left(t\right)y^{\text{'}}\left(t\right)z\left(t\right)+x\left(t\right)y\left(t\right)z^{\text{'}}\left(t\right)$.

The required relationship between the rate of change of the surface area of the rectangular box and the rate of change of $x,y$ and $z$ of the rectangular box is $S^{\text{'}}\left(t\right)=2\left[x^{\text{'}}\left(t\right)y\left(t\right)+x\left(t\right)y^{\text{'}}\left(t\right)+y^{\text{'}}\left(t\right)z\left(t\right)+y\left(t\right)z^{\text{'}}\left(t\right)+z^{\text{'}}\left(t\right)x\left(t\right)+z\left(t\right)x^{\text{'}}\left(t\right)\right]$

Step 1. Given information and formulas

The volume of rectangular box is $V=xyz$

The surface area of rectangular box is $S=2\left(xy+yz+zx\right)$

Also $V=V\left(t\right)$, $x=x\left(t\right)$, $y=y\left(t\right)$ and $z=z\left(t\right)$.

Step 2. Explanation of part a

Now $V\left(t\right)=x\left(t\right)\times y\left(t\right)\times z\left(t\right)$

Differentiate both sides with respect to $t$

$\Rightarrow V^{\text{'}}\left(t\right)=x^{\text{'}}\left(t\right)y\left(t\right)z\left(t\right)+x\left(t\right)y^{\text{'}}\left(t\right)z\left(t\right)+x\left(t\right)y\left(t\right)z^{\text{'}}\left(t\right)$

Step 3. Explanation of part b

Now $S\left(t\right)=2\left[x\left(t\right)y\left(t\right)+y\left(t\right)z\left(t\right)+z\left(t\right)x\left(t\right)\right]$

Differentiate both sides with respect to $t$

$\Rightarrow S^{\text{'}}\left(t\right)=2\left[x^{\text{'}}\left(t\right)y\left(t\right)+x\left(t\right)y^{\text{'}}\left(t\right)+y^{\text{'}}\left(t\right)z\left(t\right)+y\left(t\right)z^{\text{'}}\left(t\right)+z^{\text{'}}\left(t\right)x\left(t\right)+z\left(t\right)x^{\text{'}}\left(t\right)\right]$