Problem 41 - Thomas Calculus

Question

A bead is formed from a sphere of radius 5 by drilling through a diameter of the sphere with a drill bit of radius \(3 .\) a. Find the volume of the bead. b. Find the volume of the removed portion of the sphere.

Step-by-Step Solution

Answer

a. The volume of the bead is \(42\pi\). b. The volume of the removed portion is \(\frac{374\pi}{3}\).

Step 1: Calculate the Volume of the Original Sphere

The volume of a sphere is given by the formula \( V = \frac{4}{3} \pi r^3 \). For the original sphere with radius 5, the volume is \( V = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi \times 125 = \frac{500}{3} \pi \).

Step 2: Calculate the Volume of the Cylinder Removed

The volume of a cylinder is given by the formula \( V = \pi r^2 h \), where \( r \) is the radius of the base and \( h \) is the height. Since the cylinder is drilled through the sphere, it spans the diameter of the sphere as its height. Thus, the height \( h \) is 10. The radius of the cylinder is 3, so its volume is \( V = \pi (3)^2 (10) = 90 \pi \).

Step 3: Calculate the Volume of the Spherical Caps

The cylinder removes more material than just the cylindrical portion; we have to subtract the spherical caps at each end. The volume of each spherical cap is given by \( V = \frac{\pi h^2}{3} (3R - h) \), where \( R = 5 \), \( r = 3 \), and \( h = 2 \). Thus, the volume of one cap is \( \frac{\pi (2^2)}{3}(3 \times 5 - 2) = \frac{4\pi}{3}(13) = \frac{52\pi}{3} \). There are two caps, so their total volume is \( 2 \times \frac{52\pi}{3} = \frac{104\pi}{3} \).

Step 4: Calculate the Volume of the Removed Portion

The total volume removed by the drill is the volume of the cylinder plus the volume of the two spherical caps. So, the total removed volume \( V = 90\pi + \frac{104\pi}{3} = \frac{270\pi}{3} + \frac{104\pi}{3} = \frac{374\pi}{3} \).

Step 5: Calculate the Volume of the Bead

The volume of the bead is the volume of the original sphere minus the volume of the removed portion. Thus, \( V = \frac{500\pi}{3} - \frac{374\pi}{3} = \frac{126\pi}{3} = 42\pi \).

Key Concepts

Sphere Volume Calculation

The volume of a sphere is a fundamental concept in geometry. It's calculated using the formula \[ V = \frac{4}{3} \pi r^3 \] where \( V \) represents the volume and \( r \) is the radius. This formula tells us how much space is inside the sphere.

To understand this better with an example, let’s consider a sphere with a radius of 5 units. Plugging the radius into the formula gives us:

• Calculate \( r^3 \): \( 5^3 = 125 \)
• Substitute into the formula: \( V = \frac{4}{3} \pi \times 125 \)
• Simplify to find the volume: \( V = \frac{500}{3} \pi \)

This shows there are exactly \( \frac{500}{3} \pi \) cubic units inside a sphere with a radius of 5. Understanding this simple yet powerful formula allows you to determine the volume of any sphere, just by knowing its radius.

Whenever you're working on problems involving spheres, start by identifying the radius, then apply the formula. It's as simple as that!

Cylinder Volume Calculation

Calculating the volume of a cylinder is straightforward once you know the formula: \[ V = \pi r^2 h \] Here, \( V \) is the volume, \( r \) is the radius of the base, and \( h \) is the height of the cylinder. The cylinder's shape is like a can, and its volume measures how much it can hold inside.

Consider a cylinder drilled through a sphere along its diameter. If this cylinder has a base radius of 3 and a height equal to the sphere's diameter (10 in this example), we calculate the volume as follows:

• Square the radius: \( r^2 = 3^2 = 9 \)
• Plug in values: \( V = \pi \times 9 \times 10 \)
• Simplify: \( V = 90 \pi \)

So, the cylinder's volume is \( 90 \pi \) cubic units. This method can be used to find the volume of any cylinder once you know its radius and height.

Make sure to always use the same units for radius and height to maintain consistency in your volume units. It’s easy once you practice a couple of times!

Spherical Caps Volume

Finding the volume of spherical caps can initially seem complex, but breaking it down simplifies things. A spherical cap is like a dome cut from a sphere. To find its volume, use:\[ V = \frac{\pi h^2}{3} (3R - h) \] where \( V \) is the volume, \( h \) is the height of the cap, and \( R \) is the sphere's radius.

Imagine drilling a hole through a sphere, leaving behind spherical caps. For each cap, let’s say:

• \( h = 2 \)
• \( R = 5 \)

Use the formula:

• Plug in: \( V = \frac{\pi (2^2)}{3} (3 \times 5 - 2) \)
• Calculate: \( \frac{4\pi}{3} \times 13 \)
• Result: \( \frac{52\pi}{3} \) cubic units per cap

With two caps, the total volume becomes \( 2 \times \frac{52\pi}{3} = \frac{104\pi}{3} \).

By understanding this, you can manage any problem involving spherical caps. Recognizing the pattern in the formula helps quickly estimate the volume.