A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. (Hint: Surface area=4πr².)

검증된 단계별 안내

Start by identifying the given information: the rate of change of the volume of the snowball is proportional to its surface area. The surface area of a sphere is given by \( A = 4\pi r^2 \).

Express the volume \( V \) of the sphere in terms of its radius \( r \) using the formula \( V = \frac{4}{3}\pi r^3 \).

Differentiate the volume \( V \) with respect to time \( t \) to find \( \frac{dV}{dt} \). Using the chain rule, \( \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt} \).

Since the rate of change of the volume is proportional to the surface area, we have \( \frac{dV}{dt} = k \cdot 4\pi r^2 \), where \( k \) is a constant of proportionality.

Equate the two expressions for \( \frac{dV}{dt} \): \( 4\pi r^2 \frac{dr}{dt} = k \cdot 4\pi r^2 \). Simplify to find \( \frac{dr}{dt} = k \), showing that the rate of change of the radius is constant.

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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 given by the formula A = 4πr², where r is the radius. This formula indicates that the surface area increases with the square of the radius. Understanding this relationship is crucial because the problem states that the rate of melting is proportional to this surface area, linking the geometry of the sphere to its rate of change.

Rate of Change

In calculus, the rate of change refers to how a quantity changes in relation to another variable. In this context, we are interested in how the radius of the snowball changes over time as it melts. By establishing a relationship between the surface area and the radius, we can derive the rate of change of the radius with respect to time.

Proportional Relationships

A proportional relationship means that one quantity changes at a constant rate relative to another. In this scenario, the rate at which the snowball melts is proportional to its surface area. This implies that as the surface area decreases, the radius will also change at a consistent rate, leading to the conclusion that the rate of change of the radius remains constant.