Problem 57 - Calculus of a Single Variable

Question

Manufacturing A manufacturer drills a hole through the center of a metal sphere of radius \(R\) . The hole has a radius \(r .\) Find the volume of the resulting ring.

Step-by-Step Solution

Answer

The volume of the ring left after drilling a hole of radius \(r\) within a sphere of radius \(R\) is given by \( \text{Volume of the ring} = \text{Volume of the sphere} - \text{Volume of the cylinder} \).

Step 1: Find the volume of the sphere

We start by finding the volume of the sphere using the formula for the volume of a sphere, \(V = \frac{4}{3}\pi R^3\)

Step 2: Find the volume of the cylinder

Next, we calculate the volume of the cylinder using its formula \(V = \pi r^2 h\), where \(r\) is the radius of the cylinder and \(h\) the height, which is the same as the diameter of the sphere: \(2R\).

Step 3: Subtract the volume of the cylinder from the sphere

Finally, we subtract the volume of the cylinder from that of the sphere to get the volume of the metal ring left: \( \text{Volume of the ring} = \text{Volume of the sphere} - \text{Volume of the cylinder} \)

Key Concepts

volume of a sphere

To comprehend the problem, you first need to understand the concept of a sphere's volume. A sphere is a perfectly round geometrical object in three-dimensional space. Imagine a basketball, perfect and round from every angle you look. The formula to calculate the volume of a sphere is:

• \( V = \frac{4}{3}\pi R^3 \)

where:

• \(V\) = volume of the sphere,
• \(R\) = radius of the sphere.

This formula lets you calculate how much space an entire sphere occupies. It's a crucial formula in calculus applications, especially when you're determining volumes of objects formed by revolution.

cylinder volume formula

Next, it's time to consider how to calculate the volume of a cylinder. In this scenario, imagine a can of soda, tall and cylindrical. The formula for the volume of a cylinder is:

• \( V = \pi r^2 h \)

where:

• \(V\) = volume of the cylinder,
• \(r\) = radius of the base of the cylinder,
• \(h\) = height of the cylinder.

This formula provides the space taken by the cylindrical shape, equivalent to a solid drill passing through an object. In the manufacturing process of our problem, the height \(h\) is equivalent to the sphere's diameter \(2R\), bridging two shapes into one unified solution.

solid of revolution

The concept of a solid of revolution is essential when working with shapes like spheres and cylinders together. A solid of revolution is simply a three-dimensional object created by rotating a two-dimensional shape around an axis.
Imagine taking a flat disk and spinning it around its center. This motion generates a solid shape, a technique often used in calculus applications for finding volumes.

• The sphere can be considered as a circle rotated about its diameter.
• The cylinder is considered as a rectangle rotated around its horizontal axis.

Combining these shapes through techniques such as drilling showcases the versatility of calculus in real-life problem solving and manufacturing processes.

manufacturing processes

In manufacturing, understanding these geometrical concepts is crucial. Many parts are created by modifying raw materials into specific shapes. The alteration often involves subtracting material, such as drilling a hole.

• By starting with a sphere and removing a cylinder, you craft the remaining shape.
• The precision of these calculations ensures that the final product meets required specifications.

Calculations like understanding the volume of a sphere and that of a cylinder enable engineers to optimize technology with minimal waste and maximum efficiency.
As we solve geometrical issues, it emphasizes why calculus applications are indispensable in modern manufacturing processes.