Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.6.39a

Chapter 6, Problem 6.6.39a

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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