A spherical balloon is inflated and its volume increases at a rate of 15 in³/min. What is the rate of change of its radius when the radius is 10 in?

검증된 단계별 안내

Start by recalling the formula for the volume of a sphere: V = (4/3)πr³, where V is the volume and r is the radius.

Differentiate the volume formula with respect to time t to find the relationship between the rate of change of volume and the rate of change of radius. This gives us: dV/dt = 4πr²(dr/dt).

Substitute the given rate of change of volume, dV/dt = 15 in³/min, into the differentiated equation: 15 = 4π(10)²(dr/dt).

Solve for dr/dt, the rate of change of the radius, by isolating dr/dt in the equation: dr/dt = 15 / (4π(10)²).

Simplify the expression to find the rate of change of the radius when the radius is 10 inches. This will give you the value of dr/dt in inches per minute.

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주요 개념

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

Volume of a Sphere

The volume of a sphere is given by the formula V = (4/3)πr³, where V is the volume and r is the radius. This relationship shows how the volume changes with respect to the radius, which is crucial for understanding how the volume increase affects the radius.

Related Rates

Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to relate the rate of change of the volume of the balloon to the rate of change of its radius, using derivatives to connect these rates.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. In this context, it allows us to express the derivative of the volume with respect to time in terms of the derivative of the radius with respect to time, facilitating the calculation of the radius's rate of change.