Solving for the Rate of Radius Increase

Oct 13, 2025 by windailys.com 32 views

Introduction

In this article, we will explore the relationship between the volume of a sphere and its radius, and use this relationship to calculate the rate at which the radius of a spherical balloon is increasing at a given instant. We will use the concept of related rates, a technique in calculus that allows us to find the rate at which a quantity is changing by relating it to another quantity that is changing.

The Volume of a Sphere

The volume of a sphere is given by the equation:

V = \frac{4}{3}\pi r^3

where V is the volume of the sphere and r is its radius.

Related Rates

We are given that the balloon is inflating with helium at a rate of $144 \pi \frac{ft^3}{min}$ . This means that the volume of the balloon is increasing at a rate of $144 \pi \frac{ft^3}{min}$ . We want to find the rate at which the radius of the balloon is increasing at the instant the radius is 3 ft.

To do this, we need to relate the rate of change of the volume to the rate of change of the radius. We can do this by differentiating the equation for the volume of a sphere with respect to time:

\frac{dV}{dt} = \frac{d}{dt} \left(\frac{4}{3}\pi r^3\right)

Using the chain rule, we get:

\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}

Solving for the Rate of Radius Increase

We are given that the rate of change of the volume is $144 \pi \frac{ft^3}{min}$ , and we want to find the rate of change of the radius when the radius is 3 ft. We can substitute these values into the equation above:

144 \pi = 4\pi (3)^2 \frac{dr}{dt}

Simplifying, we get:

144 \pi = 36\pi \frac{dr}{dt}

Dividing both sides by $36\pi$ , we get:

\frac{dr}{dt} = \frac{144 \pi}{36 \pi}

\frac{dr}{dt} = 4

Therefore, the rate at which the radius of the balloon is increasing at the instant the radius is 3 ft is 4 ft/min.

Conclusion

In this article, we used the concept of related rates to calculate the rate at which the radius of a spherical balloon is increasing at a given instant. We related the rate of change of the volume to the rate of change of the radius, and used this relationship to find the rate of radius increase when the radius is 3 ft. This technique is useful in a wide range of applications, from physics and engineering to economics and finance.

Mathematical Derivations

Derivation of the Equation for the Volume of a Sphere

The volume of a sphere is given by the equation:

V = \frac{4}{3}\pi r^3

To derive this equation, we can use the method of cylindrical shells. Imagine a sphere with radius r and height h. We can divide the sphere into thin cylindrical shells, each with radius r and height dh. The volume of each shell is given by:

dV = \pi r^2 dh

We can integrate this expression over the entire sphere to get the total volume:

V = \int_0^r \pi r^2 dh

Evaluating the integral, we get:

V = \frac{4}{3}\pi r^3

Derivation of the Equation for the Rate of Change of the Volume

We are given that the rate of change of the volume is $144 \pi \frac{ft^3}{min}$ . We want to find the rate of change of the radius when the radius is 3 ft. We can relate the rate of change of the volume to the rate of change of the radius by differentiating the equation for the volume of a sphere with respect to time:

\frac{dV}{dt} = \frac{d}{dt} \left(\frac{4}{3}\pi r^3\right)

Using the chain rule, we get:

\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}

We can substitute the given values into this equation to solve for the rate of radius increase.

Derivation of the Equation for the Rate of Change of the Radius

144 \pi = 4\pi (3)^2 \frac{dr}{dt}

Simplifying, we get:

144 \pi = 36\pi \frac{dr}{dt}

Dividing both sides by $36\pi$ , we get:

\frac{dr}{dt} = \frac{144 \pi}{36 \pi}

\frac{dr}{dt} = 4

Q&A: Related Rates and Spherical Balloons

Q: What is related rates?

A: Related rates is a technique in calculus that allows us to find the rate at which a quantity is changing by relating it to another quantity that is changing. In the context of the spherical balloon, we used related rates to find the rate at which the radius of the balloon is increasing at a given instant.

Q: How do I apply related rates to a problem?

A: To apply related rates, you need to:

Identify the quantities that are changing and the relationships between them.
Differentiate the equation that relates the quantities with respect to time.
Substitute the given values into the differentiated equation and solve for the unknown quantity.

Q: What is the formula for the volume of a sphere?

A: The formula for the volume of a sphere is:

V = \frac{4}{3}\pi r^3

where V is the volume of the sphere and r is its radius.

Q: How do I find the rate of change of the radius of a sphere?

A: To find the rate of change of the radius of a sphere, you need to:

Differentiate the equation for the volume of a sphere with respect to time.
Substitute the given values into the differentiated equation and solve for the rate of change of the radius.

Q: What is the rate of change of the radius of the balloon when the radius is 3 ft?

A: The rate of change of the radius of the balloon when the radius is 3 ft is 4 ft/min.

Q: How do I use related rates to solve problems in physics and engineering?

A: Related rates can be used to solve a wide range of problems in physics and engineering, including:

Finding the rate at which a quantity is changing in a system.
Determining the rate at which a system is changing in response to external forces.
Analyzing the behavior of complex systems and predicting their behavior over time.

Q: What are some common applications of related rates?

A: Some common applications of related rates include:

Calculating the rate at which a population is growing or declining.
Determining the rate at which a chemical reaction is occurring.
Analyzing the behavior of complex systems, such as electrical circuits or mechanical systems.

Q: How do I practice related rates?

A: To practice related rates, you can try solving problems that involve related rates, such as:

Finding the rate at which a quantity is changing in a system.
Determining the rate at which a system is changing in response to external forces.
Analyzing the behavior of complex systems and predicting their behavior over time.

You can also try using online resources, such as calculus textbooks or online calculators, to practice related rates.

Conclusion

In this article, we used related rates to calculate the rate at which the radius of a spherical balloon is increasing at a given instant. We also answered some common questions about related rates and provided some tips for practicing related rates. We hope this article has been helpful in understanding related rates and how to apply them to solve problems in physics and engineering.