A spherical balloon is filled with 4500 πcubic meters of helium gas. If a leak in the balloon ca

step 1 We know volume equal volume of square equals 4x3 pi r cube. After 49 minute, volume equal 4 ,500

A spherical balloon is filled with 4500 πcubic meters of...

A spherical balloon is filled with $4500 \pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72 \pi$ cubic meters per minute, then the rate (in meters per minute) at which theradius of the balloon decreases 49 minutes after the leakage began is (A) $\frac{9}{7}$ (B) $\frac{7}{9}$ (C) $\frac{2}{9}$ (D) $\frac{9}{2}$

A spherical balloon is filled with $4500 \pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72 \pi$ cubic meters per minute, then the rate (in meters per minute) at which theradius of the balloon decreases 49 minutes after the leakage began is
(A) $\frac{9}{7}$
(B) $\frac{7}{9}$
(C) $\frac{2}{9}$
(D) $\frac{9}{2}$

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Key Concepts

Volume of a Sphere

This concept involves the formula for the volume of a sphere, V = (4/3)?r³, which relates the volume of a sphere to its radius. It is used to connect geometric properties with the dimensions of the sphere, forming the basis for translating volumetric changes into changes in radius.

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to time by relating it to the rate of change of another quantity. In these problems, variables that depend on time are connected through an equation, and their time derivatives are related by differentiating that equation.

Chain Rule in Differentiation

The chain rule is a fundamental calculus tool used for finding the derivative of a composite function. When differentiating equations like V = (4/3)?r³, where the radius r is itself a function of time, the chain rule enables the connection between dV/dt (the rate of change of volume) and dr/dt (the rate of change of radius).

Algebraic Manipulation in Rate Problems

Once an equation relating the rates is obtained (for example, through differentiation), algebraic manipulation is used to isolate the desired quantity, such as dr/dt. This involves substituting known values and solving the resulting equation to find the unknown rate.

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