Melting Snowball Calculus Problem: A Step-by-Step Solution

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Introduction

This article delves into a classic calculus problem involving rates of change, specifically focusing on a melting spherical snowball. We will explore how the snowball's volume decreases as its radius shrinks, applying the principles of related rates. This type of problem highlights the power of calculus in describing real-world phenomena and provides a practical application of differentiation. Understanding related rates is crucial for various fields, including physics, engineering, and even economics, where quantities are often interconnected and change over time. This detailed exploration will not only clarify the solution to the specific problem but also enhance your understanding of the underlying concepts and problem-solving techniques applicable to similar scenarios. By the end of this article, you'll have a solid grasp of how to tackle related rates problems involving spheres and other geometric shapes. This problem serves as a perfect example to illustrate the relationship between the rate of change of the radius and the rate of change of the volume of a sphere. The key to solving this problem lies in understanding the formula for the volume of a sphere and applying the chain rule of differentiation. This allows us to connect the rate at which the radius is decreasing to the rate at which the volume is decreasing. We will also discuss the importance of interpreting the negative sign in the context of the problem, ensuring that the final answer represents the magnitude of the volume decrease.

Problem Statement

A spherical snowball is melting in such a way that its radius is decreasing at a rate of 0.4 cm/min. At what rate is the volume of the snowball decreasing when the radius is 18 cm? (Note: The answer should be a positive number).

Solution

1. Identify Variables and Given Information

First, let's clearly define the variables involved in this problem. Let:

  • V represent the volume of the snowball (in cm³).
  • r represent the radius of the snowball (in cm).
  • t represent time (in minutes).

We are given the following information:

  • The rate at which the radius is decreasing: dr/dt = -0.4 cm/min (negative because the radius is decreasing).
  • The radius at the specific moment we're interested in: r = 18 cm.

Our goal is to find the rate at which the volume is decreasing, which is dV/dt, when r = 18 cm. This is a classic related rates problem, where we are given the rate of change of one variable and asked to find the rate of change of another variable that is related to the first. The key to solving these problems is to find an equation that relates the variables and then differentiate that equation with respect to time.

2. Establish the Relationship Between Variables

The volume of a sphere is given by the formula:

V = (4/3)πr³

This equation establishes the relationship between the volume V and the radius r of the snowball. This is the fundamental equation that we will use to solve the problem. Understanding the geometry of the situation and knowing the relevant formulas is crucial in setting up the problem correctly. In this case, the snowball is a sphere, so we need to use the formula for the volume of a sphere. This formula tells us that the volume is directly proportional to the cube of the radius, which means that even a small change in the radius can lead to a significant change in the volume.

3. Differentiate with Respect to Time

Now, we need to differentiate both sides of the volume equation with respect to time t. This is where the chain rule of calculus comes into play. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, times the derivative of the inner function. In this case, we are differentiating V with respect to t, and V is a function of r, which is itself a function of t. Applying the chain rule, we get:

dV/dt = (4/3)π * 3r² * (dr/dt)

Simplifying the equation, we have:

dV/dt = 4πr² (dr/dt)

This equation now relates the rate of change of the volume (dV/dt) to the rate of change of the radius (dr/dt). This is the crucial step in solving the problem, as it allows us to connect the given information to the quantity we want to find. The chain rule is a fundamental concept in calculus, and it is essential for solving related rates problems. It allows us to differentiate composite functions, which are functions that are made up of other functions.

4. Substitute Known Values

We know that dr/dt = -0.4 cm/min and r = 18 cm. Substitute these values into the equation:

dV/dt = 4π(18 cm)² (-0.4 cm/min)

5. Calculate the Result

Now, we perform the calculation:

dV/dt = 4π(324 cm²) (-0.4 cm/min)

dV/dt = -518.4π cm³/min

6. Interpret the Result

The rate at which the volume is decreasing is -518.4π cm³/min. The negative sign indicates that the volume is decreasing. However, the problem asks for the rate at which the volume is decreasing as a positive number. Therefore, the answer is the absolute value of our result:

|dV/dt| = 518.4π cm³/min

Approximating π as 3.14159, we get:

|dV/dt| ≈ 1628.6 cm³/min

Therefore, the volume of the snowball is decreasing at a rate of approximately 1628.6 cm³/min when the radius is 18 cm. Understanding the physical meaning of the result is crucial. In this case, the negative sign indicates that the volume is decreasing. However, the problem asks for the rate at which the volume is decreasing as a positive number, so we take the absolute value of our result.

Conclusion

We have successfully determined the rate at which the volume of the snowball is decreasing using the principles of related rates. The key steps involved identifying the variables, establishing the relationship between them using the formula for the volume of a sphere, differentiating with respect to time using the chain rule, substituting the known values, and interpreting the result in the context of the problem. This problem illustrates the practical application of calculus in understanding dynamic processes and provides a valuable example for tackling similar problems involving rates of change. Related rates problems are a fundamental topic in calculus, and they have applications in various fields. By understanding the concepts and techniques involved in solving these problems, you can gain a deeper appreciation for the power of calculus in describing real-world phenomena. This problem also highlights the importance of careful attention to detail, especially when dealing with units and signs. A small error in the calculation or interpretation can lead to a significantly different result. Therefore, it is essential to double-check your work and make sure that your answer makes sense in the context of the problem. The application of calculus, specifically the concept of related rates, allowed us to quantify this change. This example underscores the importance of calculus in modeling and understanding dynamic processes in the real world. Whether it's the melting of a snowball, the flow of fluids, or the movement of objects, calculus provides the tools to analyze and predict change. The rate at which the volume is decreasing depends on both the current radius of the snowball and the rate at which the radius is decreasing. This means that the volume decreases more rapidly when the snowball is larger. This makes intuitive sense because the surface area of a larger sphere is greater, so more of the snowball is exposed to the melting process.

Keywords

Spherical snowball melting, rate of change, calculus, related rates, volume of sphere, differentiation, chain rule, decreasing rate, radius, problem-solving, mathematics.

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Melting Snowball Calculus Problem How to Solve Related Rates