Understanding the Cone Volume Problem

This problem asks us to find the diameter of the base of a cone, given its volume and its height. A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.

We are provided with the following information:

• Volume of the cone (\(V\)) = 73920 cm³
• Height of the cone (\(h\)) = 160 cm

Our goal is to determine the diameter (\(d\)) of the circular base.

Cone Volume Formula Explained

The volume of a cone is calculated using a standard formula that relates its height and the radius of its base. The formula is:

\(V = \frac{1}{3} \pi r^2 h\)

Where:

• \(V\) is the volume of the cone
• \(\pi\) (pi) is a mathematical constant, approximately equal to 3.14159 or \(\frac{22}{7}\)
• \(r\) is the radius of the circular base
• \(h\) is the height of the cone

The diameter (\(d\)) of the base is twice the radius (\(r\)), so \(d = 2r\).

Step-by-Step Calculation of Cone Radius

To find the diameter, we first need to find the radius (\(r\)) of the base using the given volume and height. We can rearrange the volume formula to solve for \(r^2\):

\(V = \frac{1}{3} \pi r^2 h\)

Multiply both sides by 3:

\(3V = \pi r^2 h\)

Divide both sides by \(\pi h\):

\(r^2 = \frac{3V}{\pi h}\)

Now, substitute the given values for \(V\) and \(h\) into this formula. We will use \(\pi = \frac{22}{7}\) for the calculation.

\(r^2 = \frac{3 \times 73920}{\frac{22}{7} \times 160}\)

To simplify the calculation, we can rewrite the denominator:

\(r^2 = \frac{3 \times 73920}{\frac{22 \times 160}{7}}\)

Multiply the numerator by the reciprocal of the denominator:

\(r^2 = \frac{3 \times 73920 \times 7}{22 \times 160}\)

Calculate the values:

\(3 \times 73920 = 221760\)

\(22 \times 160 = 3520\)

So, the equation becomes:

\(r^2 = \frac{221760 \times 7}{3520}\)

Now, perform the multiplication in the numerator:

\(r^2 = \frac{1552320}{3520}\)

Perform the division:

\(r^2 = 441\)

To find \(r\), take the square root of both sides:

\(r = \sqrt{441}\)

\(r = 21 \text{ cm}\)

The radius of the base is 21 cm.

Calculating the Cone Base Diameter

The diameter of the base is simply twice the radius.

\(d = 2r\)

Substitute the calculated radius value:

\(d = 2 \times 21 \text{ cm}\)

\(d = 42 \text{ cm}\)

Thus, the diameter of the base of the cone is 42 cm.

Revision Table: Cone Geometry Formulas

Additional Information on Cones

Cones are fascinating geometric shapes. Here are a few more key terms and concepts related to cones:

• Slant Height (\(l\)): The distance from the apex to any point on the circumference of the base. In a right cone (where the apex is directly above the center of the base), the height, radius, and slant height are related by the Pythagorean theorem: \(l^2 = r^2 + h^2\).
• Surface Area: A cone has two surfaces: the base and the lateral surface.
  • Area of the base = \(\pi r^2\)
  • Lateral surface area = \(\pi r l\)
  • Total surface area = Area of base + Lateral surface area = \(\pi r^2 + \pi r l = \pi r (r + l)\)
• Types of Cones: The cone discussed here is a right circular cone. Oblique cones have the apex not directly above the center of the base.

Understanding these concepts helps in solving various problems involving cones, such as finding surface area or slant height.