Finding the Height of a Cone from Curved Surface Area and Radius

This problem asks us to find the height of a cone when we are given its radius and its curved surface area (CSA).

Given Information

• Radius of the cone (\(r\)) = 14 cm
• Curved Surface Area of the cone (CSA) = 2200 cm²

Formula for Curved Surface Area of a Cone

The curved surface area of a cone is calculated using the formula:

\(\text{CSA} = \pi r l\)

where:

• \(r\) is the radius of the base
• \(l\) is the slant height of the cone
• \(\pi\) is a mathematical constant (we usually use \(\frac{22}{7}\) for calculations involving multiples of 7)

Calculating the Slant Height (\(l\))

We can use the given CSA and radius to find the slant height (\(l\)). We have:

\(2200 = \frac{22}{7} \times 14 \times l\)

Simplify the equation:

\(2200 = 22 \times 2 \times l\)

\(2200 = 44 \times l\)

Now, solve for \(l\):

\(l = \frac{2200}{44}\)

\(l = 50 \text{ cm}\)

So, the slant height of the cone is 50 cm.

Relationship Between Radius, Height, and Slant Height

The radius (\(r\)), height (\(h\)), and slant height (\(l\)) of a cone form a right-angled triangle, with the slant height being the hypotenuse. According to the Pythagorean theorem, the relationship is:

\(l^2 = r^2 + h^2\)

Calculating the Height (\(h\))

We can rearrange the Pythagorean theorem formula to solve for the height (\(h\)):

\(h^2 = l^2 - r^2\)

\(h = \sqrt{l^2 - r^2}\)

Now, substitute the values of \(l\) (50 cm) and \(r\) (14 cm) into this formula:

\(h = \sqrt{(50)^2 - (14)^2}\)

\(h = \sqrt{2500 - 196}\)

\(h = \sqrt{2304}\)

To find the square root of 2304, we can use prime factorization or recall common squares. Let's find the square root:

\(h = 48 \text{ cm}\)

The height of the cone is 48 cm.

Final Answer

The height of the cone is 48 cm.

Revision Table: Cone Formulas and Properties

Additional Information: Understanding Cone Dimensions

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. The key dimensions of a cone are:

• Radius (r): The distance from the center of the circular base to any point on its edge.
• Height (h): The perpendicular distance from the apex to the center of the base. This is also called the altitude.
• Slant Height (l): The distance from the apex to any point on the circumference of the base. This lies along the curved surface.

These three dimensions are related by the Pythagorean theorem because the height, radius, and slant height form a right-angled triangle with the height and radius as the perpendicular sides and the slant height as the hypotenuse, provided the cone is a right circular cone (where the apex is directly above the center of the base).

When solving problems involving cones, it's crucial to identify which dimensions are given and which formula is needed (CSA, TSA, Volume, or finding a missing dimension using the Pythagorean theorem).