Problem 59 - College Algebra with Modeling and Visualization

Question

Approximate, to the nearest tenth, the volume and surface area (side only) of the cone satisfying the given conditions, where \(r\) is the radius and \(\mathrm{h}\) is the height. Three times \(r\) equals \(h,\) and \(r=2.4\) feet

Step-by-Step Solution

Answer

Volume: 136.8 cubic feet; Surface Area: 57.3 square feet.

Step 1: Understand the Problem

We need to find the volume and the lateral surface area of a cone. Given: radius \( r = 2.4 \) feet and the height \( h = 3 \times r \).

Step 2: Calculate the Height

Substitute \( r = 2.4 \) into the equation \( h = 3 \times r \): \[h = 3 \times 2.4 = 7.2 \text{ feet.}\]

Step 3: Calculate the Volume of the Cone

The formula for the volume \( V \) of a cone is \( V = \frac{1}{3} \pi r^2 h \).Substitute \( r = 2.4 \) and \( h = 7.2 \): \[V = \frac{1}{3} \pi (2.4)^2 (7.2) \]Calculate this:\[V \approx \frac{1}{3} \pi \times 5.76 \times 7.2 = \frac{1}{3} \times 130.6368 \approx 43.5456 \pi \approx 136.8 \]Thus, the volume is approximately \( 136.8 \text{ cubic feet.} \)

Step 4: Calculate the Lateral Surface Area

The lateral surface area \( A \) of a cone is given by \( A = \pi r l \), where \( l \) is the slant height.First, find the slant height \( l \) using the Pythagorean theorem: \[l = \sqrt{r^2 + h^2} = \sqrt{2.4^2 + 7.2^2} \approx \sqrt{5.76 + 51.84} = \sqrt{57.6} \approx 7.6 \text{ feet.} \]Now calculate \( A \):\[A = \pi \times 2.4 \times 7.6 \approx 57.29 \]So, the lateral surface area is approximately \( 57.3 \text{ square feet.} \)

Step 5: Conclusion

The volume of the cone is approximately \( 136.8 \) cubic feet, and the lateral surface area is approximately \( 57.3 \) square feet.

Key Concepts

Volume of a Cone

The volume of a cone is determined by the formula \( V = \frac{1}{3} \pi r^2 h \), where \( V \) is the volume, \( r \) is the radius of the base, and \( h \) is the height of the cone. This formula comes from the relationship between the volume of a cylinder and a cone with the same base and height.

To visualize, imagine a cylinder and a cone with identical base and height. The cone fits inside the cylinder, occupying just one-third of the space. This is intuitive because volume considers all filled space.

For our problem, with \( r = 2.4 \text{ feet} \) and \( h = 7.2 \text{ feet} \), the volume is calculated as follows:

• Square the radius: \( r^2 = (2.4)^2 = 5.76 \)
• Multiply by the height: \( 5.76 \times 7.2 = 41.472 \)
• Divide by 3 and multiply by \( \pi \): \( V = \frac{1}{3} \pi \times 41.472 \approx 136.8 \text{ cubic feet}\).

This result indicates the space that the cone would occupy if filled with a substance, like liquid or sand.

Surface Area of a Cone

The surface area of a cone is composed of two parts: the base area and the lateral (side) surface area. We're focusing on the lateral surface, which is given by \( A = \pi r l \), where \( A \) is the lateral area, \( r \) is the radius, and \( l \) is the slant height.

The lateral surface area is essentially the area of the "side" of the cone, often visualized as a sector of a circle when "unrolled." This can be easily calculated:

• First, find the slant height \( l \), which we've calculated as approximately \( 7.6 \text{ feet} \).
• Use the formula: \( A = \pi \times 2.4 \times 7.6 \)
• Compute \( A \approx 57.29 \text{ square feet} \)

This area represents the fabric or paper needed to cover the cone excluding the base.

Slant Height Calculation

Slant height is crucial for finding the cone's lateral surface area as it connects the radius and the vertical height in a right triangle. We find it using Pythagoras' theorem: \( l = \sqrt{r^2 + h^2} \).
In the context of a cone, imagine the radius and the height as legs of a right triangle, where the slant height is the hypotenuse.

For our specific exercise, the steps are:

• Calculate \( r^2 = (2.4)^2 = 5.76 \)
• Calculate \( h^2 = (7.2)^2 = 51.84 \)
• Add them: \( r^2 + h^2 = 5.76 + 51.84 = 57.6 \)
• Find the square root: \( l \approx \sqrt{57.6} \approx 7.6 \)

The slant height \( l \) allows for perfect calculations of the cone's surface and designs involving its shape.