Problem 65 - Calculus Early Transcendentals

Question

A right circular cylinder with height \(R\) and radius \(R\) has a volume of \(V_{C}=\pi R^{3}\) (height \(=\) radius). a. Find the volume of the cone that is inscribed in the cylinder with the same base as the cylinder and height \(R\). Express the volume in terms of \(V_{C}\). b. Find the volume of the hemisphere that is inscribed in the cylinder with the same base as the cylinder. Express the volume in terms of \(V_{C}\).

Step-by-Step Solution

Answer

Question: Express the volumes of an inscribed cone and hemisphere in a right circular cylinder in terms of the cylinder's volume, given that their heights and base radii are equal to the cylinder's height and radius (R). Answer: The volume of the inscribed cone is \(\frac{1}{3}V_C\), and the volume of the inscribed hemisphere is \(\frac{2}{3}V_C\), where \(V_C = \pi R^3\) is the volume of the cylinder.

Step 1: Find the volume of the inscribed cone

First, we will find the volume of the inscribed cone. The formula for the volume of a cone is given by: \(V_{cone} = \frac{1}{3}\pi r^2h\) Since the cone has the same base and height as the cylinder, its radius (r) and height (h) are also R. So, we have: \(V_{cone} = \frac{1}{3}\pi R^2R\) \(V_{cone} = \frac{1}{3}\pi R^3\)

Step 2: Express the volume of the cone in terms of \(V_C\)

We are given the volume of the cylinder as \(V_C = \pi R^3\). We can now express the volume of the cone in terms of \(V_C\): \(V_{cone} = \frac{1}{3}V_C\)

Step 3: Find the volume of the inscribed hemisphere

Now, we will find the volume of the inscribed hemisphere. The formula for the volume of a hemisphere is given by: \(V_{hemisphere} = \frac{2}{3}\pi r^3\) Since the hemisphere has the same base as the cylinder, its radius (r) is also R. So, we have: \(V_{hemisphere} = \frac{2}{3}\pi R^3\)

Step 4: Express the volume of the hemisphere in terms of \(V_C\)

We are given the volume of the cylinder as \(V_C = \pi R^3\). We can now express the volume of the hemisphere in terms of \(V_C\): \(V_{hemisphere} = \frac{2}{3}V_C\) In summary, the volume of the inscribed cone is \(\frac{1}{3}V_C\) and the volume of the inscribed hemisphere is \(\frac{2}{3}V_C\)

Key Concepts

Volume of a Cylinder

Let's start with understanding the volume of a cylinder. A cylinder is a three-dimensional shape with two parallel circular bases and a curved surface. The formula to find the volume of a cylinder is quite straightforward:\[V_{cylinder} = \pi r^2 h\]where:

• \(r\) is the radius of the base.
• \(h\) is the height.

This formula calculates how much space is inside the cylinder by multiplying the area of its base (a circle) by its height.

In our exercise, both the radius and the height are equal to \(R\). Thus, using the formula:\[V_{C} = \pi R^{3}\]We've used this setup as a base for understanding other shapes inscribed in the cylinder.

Volume of a Cone

Next, let's explore how to find the volume of a cone, especially when it's placed inside a cylinder. A cone has a circular base and tapers smoothly to a point called the apex. The formula for calculating the volume of a cone is:\[V_{cone} = \frac{1}{3} \pi r^2 h\]In this formula:

• \(r\) is the radius of the base.
• \(h\) is the height.

Notice how it's similar to that of a cylinder, but multiplied by \(1/3\) to account for the cone's tapering shape.

In our specific question, the cone shares the same base and height as the cylinder, meaning both are \(R\). Applying these values, the volume becomes:\[V_{cone} = \frac{1}{3} \pi R^2 R = \frac{1}{3} \pi R^3\]This defines the cone's volume in terms of the cylinder's volume, simplifying to\[V_{cone} = \frac{1}{3} V_{C}\]

Volume of a Hemisphere

Finally, let's discuss the volume of a hemisphere, which is half of a sphere. A hemisphere also uses a circular base like the previous shapes. To find its volume, the formula given is:\[V_{hemisphere} = \frac{2}{3} \pi r^3\]where \(r\) is the radius of the hemisphere's base.Since the hemisphere fits snugly inside the cylinder with the same radius \(R\), the calculation becomes:\[V_{hemisphere} = \frac{2}{3} \pi R^3\]This relationship can also be expressed using the volume of the cylinder:\[V_{hemisphere} = \frac{2}{3} V_{C}\]Understanding these fun geometric shapes helps in visualizing how 3D spaces are filled and gives a deeper insight into everyday objects like an ice cream cone, which may consist of an actual cone stacked with a hemisphere of ice cream on top.