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Expert-verified Found in: Page 298 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Problem Zero: Read the section and make your summary of material

the rate can be calculated in terms of the factors that change with time.

See the step by step solution

## Step 1: Given information

"RELATED RATES"

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

## Step 2: Explanation

1. A rectangular box $V=xyz\phantom{\rule{0ex}{0ex}}S=2xy+2yz+2xz$

2. Sphere $V=\frac{4}{3}\pi {r}^{3}\phantom{\rule{0ex}{0ex}}S=4\pi {r}^{2}$

3. right circular cylinder $V=\pi {r}^{2}h\phantom{\rule{0ex}{0ex}}S=2\pi rh+2\pi {r}^{2}\phantom{\rule{0ex}{0ex}}L=2\pi rh$

4. right circular cone $V=\frac{1}{3}\pi {r}^{2}h\phantom{\rule{0ex}{0ex}}S=\pi r\sqrt{{r}^{2}+{h}^{2}}+\pi {r}^{2}\phantom{\rule{0ex}{0ex}}L=\pi r\sqrt{{r}^{2}+{h}^{2}}$

## Step 3: Further calculation

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 $V=\frac{1}{3}\pi {r}^{2}h$ determines the volume. Wherever the radius and height change, the volume V changes as well.

localid="1663925684234" $V\left(t\right)=\frac{1}{3}\pi {\left[r\left(t\right)\right]}^{2}h\left(t\right)\phantom{\rule{0ex}{0ex}}\frac{dV\left(t\right)}{dt}=\frac{1}{3}\pi \left[r\left(t{\right)}^{2}\frac{dh\left(t\right)}{dt}+h\left(t\right)\frac{dr\left(t{\right)}^{2}}{dt}\right]\phantom{\rule{0ex}{0ex}}\frac{dV\left(t\right)}{dt}=\frac{1}{3}\pi \left[r\left(t{\right)}^{2}\frac{dh\left(t\right)}{dt}+h\left(t\right)r\left(t\right)\frac{dr\left(t\right)}{dt}\right]$

Thus, the rate can be measured in terms of the variables associated over time. ### Want to see more solutions like these? 