A Spherical Balloon Is Inflated With Gas At A Rate Of 500 Cubic Centimeters Per Minute.(b) How Fast Is The Radius Of The Balloon Changing At The Instant The Radius Is 90 Centimeters?

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Introduction

In this article, we will delve into a classic problem in mathematics involving the inflation of a spherical balloon. We will explore how to calculate the rate of change of the radius of the balloon at a specific instant, given the rate at which the balloon is being inflated. This problem is a great example of how to apply the concept of related rates in calculus to solve real-world problems.

The Problem

A spherical balloon is inflated with gas at a rate of 500 cubic centimeters per minute. We are asked to find how fast the radius of the balloon is changing at the instant the radius is 90 centimeters.

The Formula for the Volume of a Sphere

To solve this problem, we need to recall the formula for the volume of a sphere, which is given by:

V = (4/3)Ï€r^3

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

The Chain Rule and Related Rates

We are given the rate at which the volume of the balloon is changing, which is 500 cubic centimeters per minute. We want to find the rate at which the radius of the balloon is changing at a specific instant. To do this, we will use the chain rule and the concept of related rates.

The chain rule states that if we have a function of the form:

f(x) = g(h(x))

then the derivative of f with respect to x is given by:

f'(x) = g'(h(x)) * h'(x)

In our case, we have:

V = (4/3)Ï€r^3

We can rewrite this as:

V = g(r)

where g(r) = (4/3)Ï€r^3

We are given the rate at which the volume of the balloon is changing, which is:

dV/dt = 500

We want to find the rate at which the radius of the balloon is changing, which is:

dr/dt

Using the chain rule, we can write:

dV/dt = d(g(r))/dt

= g'(r) * dr/dt

= (4/3)Ï€(3r^2) * dr/dt

= 4Ï€r^2 * dr/dt

Solving for dr/dt

Now we have an equation that relates the rate at which the volume of the balloon is changing to the rate at which the radius of the balloon is changing. We can solve for dr/dt by dividing both sides of the equation by 4Ï€r^2:

dr/dt = (dV/dt) / (4Ï€r^2)

Substituting the Given Values

We are given that the radius of the balloon is 90 centimeters, and the rate at which the volume of the balloon is changing is 500 cubic centimeters per minute. We can substitute these values into the equation:

dr/dt = (500) / (4Ï€(90)^2)

Calculating the Value of dr/dt

Now we can calculate the value of dr/dt:

dr/dt = (500) / (4Ï€(90)^2)

= (500) / (4 * 3.14159 * 8100)

= (500) / (101 663.13)

= 0.0049

Conclusion

In this article, we used the concept of related rates in calculus to solve a problem involving the inflation of a spherical balloon. We found that the rate at which the radius of the balloon is changing at the instant the radius is 90 centimeters is approximately 0.0049 centimeters per minute.

Final Thoughts

This problem is a great example of how to apply the concept of related rates in calculus to solve real-world problems. By using the chain rule and the formula for the volume of a sphere, we were able to find the rate at which the radius of the balloon is changing at a specific instant. This type of problem is commonly encountered in fields such as physics and engineering, where it is often necessary to calculate rates of change in order to understand and predict the behavior of complex systems.

References

  • [1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
  • [2] Anton, H. (2016). Calculus: Early Transcendentals. John Wiley & Sons.
  • [3] Edwards, C. H. (2016). Calculus: Early Transcendentals. Pearson Education.

Additional Resources

Introduction

In our previous article, we explored a classic problem in mathematics involving the inflation of a spherical balloon. We used the concept of related rates in calculus to calculate the rate at which the radius of the balloon is changing at a specific instant. In this article, we will answer some common questions related to this problem.

Q&A

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

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

V = (4/3)Ï€r^3

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

Q: What is the chain rule, and how is it used in this problem?

A: The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. In this problem, we use the chain rule to relate the rate at which the volume of the balloon is changing to the rate at which the radius of the balloon is changing.

Q: How do we use the chain rule to solve this problem?

A: We start by writing the equation for the volume of the sphere:

V = (4/3)Ï€r^3

We then take the derivative of both sides of the equation with respect to time t:

dV/dt = d((4/3)Ï€r^3)/dt

Using the chain rule, we can rewrite this as:

dV/dt = (4/3)Ï€(3r^2) * dr/dt

= 4Ï€r^2 * dr/dt

Q: How do we solve for dr/dt?

A: We can solve for dr/dt by dividing both sides of the equation by 4Ï€r^2:

dr/dt = (dV/dt) / (4Ï€r^2)

Q: What is the rate at which the radius of the balloon is changing at the instant the radius is 90 centimeters?

A: We can substitute the given values into the equation:

dr/dt = (500) / (4Ï€(90)^2)

= (500) / (4 * 3.14159 * 8100)

= (500) / (101 663.13)

= 0.0049

Q: What is the significance of this problem?

A: This problem is a great example of how to apply the concept of related rates in calculus to solve real-world problems. By using the chain rule and the formula for the volume of a sphere, we were able to find the rate at which the radius of the balloon is changing at a specific instant.

Q: What are some common applications of related rates in calculus?

A: Related rates is a fundamental concept in calculus that has many applications in physics, engineering, and other fields. Some common applications include:

  • Calculating the rate at which the volume of a container is changing
  • Determining the rate at which the temperature of a substance is changing
  • Finding the rate at which the position of an object is changing

Q: What are some common mistakes to avoid when solving related rates problems?

A: Some common mistakes to avoid when solving related rates problems include:

  • Failing to use the chain rule correctly
  • Not substituting the given values into the equation
  • Not checking the units of the answer

Conclusion

In this article, we answered some common questions related to the problem of calculating the rate at which the radius of a spherical balloon is changing. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the concept of related rates in calculus.

Final Thoughts

Related rates is a fundamental concept in calculus that has many applications in physics, engineering, and other fields. By understanding how to apply the chain rule and the formula for the volume of a sphere, we can solve a wide range of problems involving related rates.

References

  • [1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
  • [2] Anton, H. (2016). Calculus: Early Transcendentals. John Wiley & Sons.
  • [3] Edwards, C. H. (2016). Calculus: Early Transcendentals. Pearson Education.

Additional Resources