Problem 55 - 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. \(\mathbf{r}=5\) centimeters, \(\mathbf{h}=6\) centimeters

Step-by-Step Solution

Answer

Volume: 157.1 cm³, Lateral Surface Area: 122.5 cm²

Step 1: Understand the Problem

We need to find both the volume and the lateral surface area of a cone with a given radius \( r = 5 \) cm and height \( h = 6 \) cm. The lateral surface area is the area of the cone excluding its base.

Step 2: Calculate the Volume of the Cone

The formula for the volume \( V \) of a cone is \( V = \frac{1}{3} \pi r^2 h \). Plugging in the given values, \( V = \frac{1}{3} \pi (5)^2 (6) \). This simplifies to \( V = 50\pi \) cm³.

Step 3: Find the Slant Height

The slant height \( l \) of a cone is calculated using the Pythagorean theorem: \( l = \sqrt{r^2 + h^2} \). Substituting the given values, \( l = \sqrt{5^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61} \approx 7.8 \) cm.

Step 4: Calculate the Lateral Surface Area

The lateral surface area \( A \) of a cone can be calculated with \( A = \pi r l \). With \( r = 5 \) cm and \( l \approx 7.8 \) cm, \( A \approx \pi (5)(7.8) \approx 39\pi \) cm².

Step 5: Round to the Nearest Tenth

To find the approximate numerical values, calculate \( 50\pi \approx 157.1 \) cm³ for the volume and \( 39\pi \approx 122.5 \) cm² for the lateral surface area. Round these to the nearest tenth.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental formula that relates the sides of a right triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides. You can write this as \( a^2 + b^2 = c^2 \). In the case of a cone, this formula helps us determine the slant height. The slant height is the hypotenuse in a right triangle where the radius \( r \) and height \( h \) of the cone represent the two other sides.
To find the slant height \( l \), you use \( l = \sqrt{r^2 + h^2} \). With our cone, where \( r = 5 \) cm and \( h = 6 \) cm, the computation is \( l = \sqrt{5^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61} \). Which approximates to \( l \approx 7.8 \) cm.
This step is critical as it provides the necessary dimension to calculate the cone's lateral surface area.

Lateral Surface Area

The lateral surface area of a cone is the area of the surface covering only the sides and not including the base. To find this, we need the slant height, which we calculated using the Pythagorean Theorem.
The formula used for the lateral surface area \( A \) is \( A = \pi r l \), where \( r \) is the radius and \( l \) is the slant height.

• With our values of \( r = 5 \) cm and \( l \approx 7.8 \) cm, the calculation becomes: \( A = \pi (5)(7.8) \approx 39\pi \text{ cm}^2 \).
• Inserting \( \pi \approx 3.14159 \) gives: \( 39\pi \approx 122.5 \text{ cm}^2 \).

The lateral surface area informs how much material you'd need to cover the cone's sides, a practical measure in real-world applications.

Volume of a Cone

The volume of a cone measures the space it occupies, computed using the formula \( V = \frac{1}{3} \pi r^2 h \). This formula efficiently calculates how much "stuff" can fit inside the cone, like water or sand.
To find the volume, substitute the given values for the radius \( r \) and height \( h \):

• \( V = \frac{1}{3} \pi (5)^2 (6) \)
• This simplifies to \( V = \frac{1}{3} \pi (25)(6) = 50\pi \text{ cm}^3 \)
• Using \( \pi \approx 3.14159 \), the approximate volume is \( 157.1 \text{ cm}^3 \).

Whether calculating for storage or construction, knowing the volume of a cone is essential for numerous practical scenarios.