What is the rate of change of the diagonal, volume, and surface of a box with dimensions a, b, and c?
I am working through some problems in my calculus workbook, and this one has me a bit stuck. If someone wouldnt mind lending a hand, I would much appreciate it.
This is the problem:
A rectangular box has dimensions a, b, and c which are changing with time. At $t = t_0$:
$a = 1 m$, $b = 2 m$, and $c = 3 m$ (m is meters).
$da/dt = db/dt = 1 m/sec$ and $dc/dt = -3 m/sec$ What is the rate of change of the volume, surface area, and the diagonals of the box at $t = t_0$.
If someone wouldnt mind lending me a hand as to the way you would solve this, I would greatly appreciate it.
Thanks Corey
$\endgroup$2 Answers
$\begingroup$Notice, the dimensions of the box $a$, $b$ & $c$ are changing w.r.t. time $t$ hence, we have
- Volume, $$V=abc$$$$\implies \frac{d V}{dt}=bc\frac{da}{dt}+ac\frac{d b}{dt}+ab\frac{dc}{dt}$$ substitute all the corresponding values, the rate of change of volume is $$\color{red}{\frac{dV}{dt}}=(2\cdot 3)(1)+(1\cdot 3)(1)+(1\cdot 2)(-3)=\color{red}{3\ m^3/s}$$
- Surface area, $$S=2(ab+bc+ac)$$$$\implies \frac{d S}{dt}=2\left(b\frac{da}{dt}+c\frac{d b}{dt}+a\frac{dc}{dt}\right)$$ substitute all the corresponding values, the rate of change of surface is $$\color{red}{\frac{dS}{dt}}=2(2\cdot 1+3\cdot 1+1\cdot (-3))=\color{red}{6\ m^2/s}$$
- Diagonal, $$D=\sqrt{a^2+b^2+c^2}\implies D^2=a^2+b^2+c^2$$ $$\implies 2D\frac{d D}{dt}=\left(2a\frac{da}{dt}+2b\frac{d b}{dt}+2c\frac{dc}{dt}\right)$$ $$\implies \frac{d D}{dt}=\frac{1}{D}\left(a\frac{da}{dt}+b\frac{d b}{dt}+c\frac{dc}{dt}\right)=\frac{1}{\sqrt{a^2+b^2+c^2}}\left(a\frac{da}{dt}+b\frac{d b}{dt}+c\frac{dc}{dt}\right)$$
substitute all the corresponding values, the rate of change of diagonal is $$\color{red}{\frac{dD}{dt}}=\frac{1}{\sqrt{1^2+2^2+3^2}}\left(1\cdot 1+2\cdot 1+3\cdot (-3)\right)=\color{red}{\frac{-6}{\sqrt {14}}\ m/s}$$
$\endgroup$ $\begingroup$Recall that the volume, surface area and diagonals are given by $$ V = abc, \quad S = 2ab+2bc+2ca, \quad D = \sqrt{a^2+b^2+c^2}. $$ I'll do the volume, and then it should be clear what to do for the others. Differentiate it as a product: $$ \frac{dV}{dt} = \frac{da}{dt}bc + a\frac{db}{dt}c + ab\frac{dc}{dt}. $$ Now just insert the values you have.
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