91Ó°ÊÓ

"RELATED RATES"

A study of the formula for the surface and area of the right-circumference cylinder, cone, sphere, and rectangular box.

1. A rectangular box

$$\begin{gathered} V=xyz \\ S=2xy+2yz+2xz \end{gathered}$$

2. Sphere

$$\begin{gathered} V=\frac{4}{3}\mathrm{Ï€}{r}^3 \\ S=4\mathrm{Ï€}{r}^2 \end{gathered}$$

3. right circular cylinder

4. right circular cone

There are two right triangle theorems.

The Pythagorean principle:$a^2+b^2=c^2$

2. The law of similar triangles: $\frac{h}{b}=\frac{II}{B},\frac{d}{b}=\frac{D}{B},\frac{d}{h}=\frac{D}{II}$

The first derivative of a function f can be used to calculate the rate of change of a function f when it is increasing or decreasing.

Example: Both the radius and the height can be used to calculate the rate of change in volume of a right circular cylinder. Since the formula $V=\frac{1}{3}\mathrm{Ï€}{r}^{2}h$determines the volume. Wherever the radius and height change, the volume V changes as well.

localid="1663925684234" $$\begin{gathered} V\left(t\right)=\frac{1}{3}\mathrm{Ï€}{\left[r\left(t\right)\right]}^{2}h\left(t\right) \\ \frac{dV\left(t\right)}{dt}=\frac{1}{3}\mathrm{Ï€}\left[r(t{)}^2\frac{dh\left(t\right)}{dt}+h(t)\frac{dr(t{)}^2}{dt}\right] \\ \frac{dV\left(t\right)}{dt}=\frac{1}{3}\mathrm{Ï€}\left[r(t{)}^2\frac{dh\left(t\right)}{dt}+h(t\left)r\right(t)\frac{dr\left(t\right)}{dt}\right] \end{gathered}$$

Thus, the rate can be measured in terms of the variables associated over time.