Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

A round hole of radius \(a\) is drilled through the center of a solid sphere of radius \(b\) (assume that \(b>a\) ). Find the volume of the solid that remains.

Step-by-Step Solution

Answer

The remaining volume is \(\frac{4}{3} \pi b^3 - 2\pi a^2 \sqrt{b^2-a^2}\).

Step 1: Understanding the Problem

We need to find the volume of a solid sphere with radius \(b\) from which a cylinder with radius \(a\) has been drilled through the center, along the diameter of the sphere.

Step 2: Calculate Volume of the Sphere

The formula for the volume of a sphere is \(V = \frac{4}{3} \pi b^3\). Substitute the radius \(b\) to find the volume of the sphere before drilling.

Step 3: Calculate Volume of the Cylinder

The volume of the cylindrical hole drilled through the sphere can be calculated using the formula \(V = \pi a^2 h\), where \(h\) is the height of the cylinder. Since the cylinder passes through the center of the sphere, \(h = 2\sqrt{b^2 - a^2}\), as determined by the geometry of the sphere and cylinder.

Step 4: Derive Expression for Cylinder's Height

To derive \(h = 2\sqrt{b^2 - a^2}\), consider the triangle formed by the radius of the sphere, the radius of the cylinder, and half of the length of the cylinder. Using the Pythagorean theorem, \((\frac{h}{2})^2 + a^2 = b^2\). Solving for \(h\) gives \(h = 2\sqrt{b^2 - a^2}\).

Step 5: Subtract Cylinder Volume from Sphere

The volume of the remaining solid is the volume of the sphere minus the volume of the cylindrical hole. Therefore, the remaining volume is \(\frac{4}{3} \pi b^3 - \pi a^2 \cdot 2\sqrt{b^2 - a^2}\).

Step 6: Conclusion

After computing the volumes, the configuration of the geometrical shapes provides the remaining volume as \(\frac{4}{3} \pi b^3 - 2\pi a^2 \sqrt{b^2 - a^2}\).

Key Concepts

Sphere

A sphere is a perfectly round 3D geometric shape, much like a basketball or a planet in space. It is defined as the set of all points in space that are at a fixed distance, known as the radius, from a central point, called the center. Since spheres are perfectly symmetrical from every angle, they are great examples when exploring volume in geometry.

The formula for the volume of a sphere is given by:\[V = \frac{4}{3} \pi r^3\]Where:

• \(V\) is the volume of the sphere.
• \( \pi \) is a constant approximately equal to 3.14159.
• \(r\) is the radius of the sphere.

This equation highlights how even small increases in the radius can greatly influence the overall volume of the sphere, due to the radius being cubed.

Cylinder

A cylinder is another essential geometric shape, characterized by its circular bases that are parallel and congruent. Imagine it as the shape of a can or a candle. Cylinders are unique in their uniform structure, with flat faces and the side shaped like a rectangle when "unwrapped".

To find the volume of a cylinder, we rely on the base area and the height of the cylinder. The formula is:\[V = \pi r^2 h\]Where:

• \(V\) represents the volume of the cylinder.
• \(r\) is the radius of the circular base.
• \(h\) is the height of the cylinder.

This means the volume depends on how large the base is and how tall the cylinder stands. Cylinders play a crucial role when calculating the volume of composite solids, like when a cylindrical hole is drilled through a sphere.

Geometric Shapes

Geometric shapes are foundational in mathematics, influencing everything from art to engineering. When considering solids like spheres and cylinders, we're delving into three-dimensional shapes.

These 3D shapes are not just abstract concepts but are motivated by real-world objects like:

• Spheres, which include balls and bubbles.
• Cylinders, such as cans and towers.

Understanding their variables like radius and height allows us to compute important properties like volume and surface area, essential for problems involving space and capacity. Recognizing the relationship between these geometric shapes justifies qualities such as the symmetry of a sphere or the structured continuity of a cylinder.

Volume Formulas

Volume formulas are equations that allow us to determine the capacity or the amount of space an object occupies. Each geometric shape has its unique formula, derived from its specific structure and symmetry.

For spheres and cylinders, these formulas are essential:

• **Sphere**: Volume \( = \frac{4}{3} \pi r^3 \) captures the entire space inside the sphere.
• **Cylinder**: Volume \( = \pi r^2 h \) gives the capacity held by the cylindrical shape.

These formulas become powerful tools when applied to complex exercises like subtracting a cylindrical volume from a sphere, as seen in scenario with a hole drilled through a solid sphere. They enable the calculation of remaining space, which aids in countless practical tasks from designing containers to understanding natural formations.