SaitechAI – Mensuration Formulas for Applications of Derivatives

Rate of change problems: differentiate mensuration formulas with respect to time \(t\).

Notation: \(\dfrac{d}{dt}\) means differentiation w.r.t. time. Variables like \(r, h, l, b, a\) are functions of \(t\).

1) 2D Figures (Area)

Shape Original Formula Differentiated Form
Square \(A=a^2\) \(\dfrac{dA}{dt}=2a\dfrac{da}{dt}\)
Rectangle \(A=lb\) \(\dfrac{dA}{dt}=l\dfrac{db}{dt}+b\dfrac{dl}{dt}\)
Circle \(A=\pi r^2\) \(\dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}\)
Triangle \(A=\frac{1}{2}bh\) \(\dfrac{dA}{dt}=\frac{1}{2}\left(b\dfrac{dh}{dt}+h\dfrac{db}{dt}\right)\)
Equilateral Triangle \(A=\frac{\sqrt{3}}{4}a^2\) \(\dfrac{dA}{dt}=\frac{\sqrt{3}}{2}a\dfrac{da}{dt}\)
Parallelogram \(A=bh\) \(\dfrac{dA}{dt}=b\dfrac{dh}{dt}+h\dfrac{db}{dt}\)

2) 3D Solids (Volume)

Solid Original Formula Differentiated Form
Cube \(V=a^3\) \(\dfrac{dV}{dt}=3a^2\dfrac{da}{dt}\)
Cuboid \(V=lbh\) \(\dfrac{dV}{dt}=bh\dfrac{dl}{dt}+lh\dfrac{db}{dt}+lb\dfrac{dh}{dt}\)
Sphere \(V=\frac{4}{3}\pi r^3\) \(\dfrac{dV}{dt}=4\pi r^2\dfrac{dr}{dt}\)
Hemisphere \(V=\frac{2}{3}\pi r^3\) \(\dfrac{dV}{dt}=2\pi r^2\dfrac{dr}{dt}\)
Cylinder \(V=\pi r^2h\) \(\dfrac{dV}{dt}=2\pi rh\dfrac{dr}{dt}+\pi r^2\dfrac{dh}{dt}\)
Cone \(V=\frac{1}{3}\pi r^2h\) \(\dfrac{dV}{dt}=\frac{1}{3}\pi\left(2rh\dfrac{dr}{dt}+r^2\dfrac{dh}{dt}\right)\)

3) Surface Area (Common in melting / evaporation)

Solid Surface Area Formula Differentiated Form
Sphere (TSA) \(S=4\pi r^2\) \(\dfrac{dS}{dt}=8\pi r\dfrac{dr}{dt}\)
Cylinder (CSA) \(S=2\pi rh\) \(\dfrac{dS}{dt}=2\pi\left(r\dfrac{dh}{dt}+h\dfrac{dr}{dt}\right)\)
Cone (CSA) \(S=\pi rl\) \(\dfrac{dS}{dt}=\pi\left(r\dfrac{dl}{dt}+l\dfrac{dr}{dt}\right)\)

4) Quick Rearrangements (Solve for a rate)

Sphere volume rate to radius rate
\[ V=\frac{4}{3}\pi r^3 \quad\Rightarrow\quad \frac{dV}{dt}=4\pi r^2\frac{dr}{dt} \] \[ \Rightarrow\quad \frac{dr}{dt}=\frac{1}{4\pi r^2}\frac{dV}{dt} \]
Cylinder volume rate
\[ V=\pi r^2h \quad\Rightarrow\quad \frac{dV}{dt}=2\pi rh\frac{dr}{dt}+\pi r^2\frac{dh}{dt} \] If \(r\) constant: \(\frac{dV}{dt}=\pi r^2\frac{dh}{dt}\). If \(h\) constant: \(\frac{dV}{dt}=2\pi rh\frac{dr}{dt}\).
Cone similarity relation (often used)
If the cone is fixed and water level forms a similar cone, then \[ \frac{r}{h}=\text{constant} \Rightarrow r=kh \] Use this to eliminate one variable before differentiating.