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Q 0.

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Calculus

Found in: Page 298

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

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.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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

$V = x y z S = 2 x y + 2 y z + 2 x z$

2. Sphere

$V = \frac{4}{3} π r^{3} S = 4 π r^{2}$

3. right circular cylinder

V = π r^{2} h S = 2 π r h + 2 π r^{2} L = 2 π r h

4. right circular cone

V = \frac{1}{3} π r^{2} h S = π r \sqrt{r^{2} + h^{2}} + π r^{2} L = π 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: $\frac{h}{b} = \frac{I I}{B}, \frac{d}{b} = \frac{D}{B}, \frac{d}{h} = \frac{D}{I I}$

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

localid="1663925684234" $V (t) = \frac{1}{3} π {[r (t)]}^{2} h (t) \frac{d V (t)}{d t} = \frac{1}{3} π [r (t)^{2} \frac{d h (t)}{d t} + h (t) \frac{d r (t)^{2}}{d t}] \frac{d V (t)}{d t} = \frac{1}{3} π [r (t)^{2} \frac{d h (t)}{d t} + h (t) r (t) \frac{d r (t)}{d t}]$

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

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