Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

9. Graphical Applications of Integrals

Introduction to Volume & Disk Method

Calculus

9. Graphical Applications of Integrals

Introduction to Volume & Disk Method

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1:40 minutes

Problem 6.6.39a

Textbook Question

In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly.

a. What is the SAV ratio of a cube with side lengths a?

Verified step by step guidance

Recall the formulas for the surface area and volume of a cube. The surface area (SA) of a cube with side length $a$ is given by $SA = 6a^{2}$ because a cube has 6 faces, each with area $a^{2}$.

The volume (V) of the cube is given by $V = a^{3}$ since volume is the product of the three side lengths, all equal to $a$.

The surface area to volume (SAV) ratio is defined as the surface area divided by the volume, so write the ratio as $\frac{SA}{V} = \frac{6a^{2}}{a^{3}}$.

Simplify the expression by dividing the powers of $a$: $\frac{6a^{2}}{a^{3}} = 6a^{2 - 3} = 6a^{-1}$.

Rewrite the simplified expression to a more standard form: $6a^{-1} = \frac{6}{a}$. This is the SAV ratio of the cube in terms of its side length $a$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Surface Area and Volume of a Cube

A cube has six equal square faces, each with area a², so its total surface area is 6a². Its volume is the cube of its side length, a³. Understanding these formulas is essential to calculate the surface area to volume (SAV) ratio.

Surface Area to Volume (SAV) Ratio

The SAV ratio compares the surface area of an object to its volume, often expressed as surface area divided by volume. It is a key concept in biology and physics, influencing processes like heat loss, as surface area affects exchange rates while volume relates to internal content.

Proportional Relationships in Biological Contexts

In biology, heat generation is proportional to volume, while heat loss is proportional to surface area. This proportionality explains why smaller animals with higher SAV ratios lose heat faster, linking mathematical ratios to real-world biological phenomena.

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In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly.

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