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.

b. What is the SAV ratio of a ball with radius a?

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Recall the formulas for the surface area and volume of a sphere (ball) with radius \(a\). The surface area \(S\) is given by \(S = 4 \pi a^2\).

The volume \(V\) of the sphere is given by \(V = \frac{4}{3} \pi a^3\).

The surface area to volume (SAV) ratio is defined as \(\frac{S}{V}\), so substitute the formulas for \(S\) and \(V\) into this ratio.

Write the SAV ratio as \(\frac{4 \pi a^2}{\frac{4}{3} \pi a^3}\) and simplify by canceling common factors such as \(4\) and \(\pi\).

After simplification, express the SAV ratio in terms of \(a\) only, which will show how the ratio depends on the radius of the ball.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Surface Area of a Sphere

The surface area of a sphere is calculated using the formula 4πa², where a is the radius. This represents the total area covering the outer layer of the ball, which is crucial for understanding how much heat can be lost through the surface.

Volume of a Sphere

The volume of a sphere is given by the formula (4/3)πa³, where a is the radius. This measures the total space inside the ball, which relates to the amount of heat generated within the object.

Surface Area to Volume (SAV) Ratio

The SAV ratio is the surface area divided by the volume, indicating how much surface is available per unit volume. For a sphere, this ratio helps explain heat retention or loss, as it compares the heat loss area to the heat-generating volume.