Problem 39 - Algebra 2

Question

A right circular cone has radius 4 inches and height 6 inches. What is the lateral area of the cone? (Lateral area of cone \(=\pi r \ell,\) where \(\ell=\) slant height)? A. 24\(\pi \mathrm{sq}\) in. B. 2\(\sqrt{13} \pi \mathrm{sq}\) in. C. 2\(\sqrt{52} \pi \mathrm{sq}\) in. D. 8\(\sqrt{13} \pi \mathrm{sq}\) in.

Step-by-Step Solution

Answer

D. 8\(\sqrt{13}\pi \mathrm{sq}\) in.

Step 1: Recall the formula for slant height

The slant height \( \ell \) of a cone can be found using the Pythagorean theorem in the right triangle formed by the radius, height, and slant height of the cone. The formula is: \( \ell = \sqrt{r^2 + h^2} \) where \( r \) is the radius and \( h \) is the height.

Step 2: Substitute the known values into the slant height formula

Given that the radius \( r = 4 \) inches and the height \( h = 6 \) inches, substitute these values into the slant height formula: \( \ell = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \).

Step 3: Calculate the slant height

Evaluate the expression \( \sqrt{52} \). The slant height \( \ell \) simplifies to \( \sqrt{52} \), which can further be simplified to \( 2\sqrt{13} \).

Step 4: Use the lateral area formula

The lateral area \( A \) of a cone is given by \( A = \pi r \ell \). Substitute \( r = 4 \) inches and \( \ell = 2\sqrt{13} \) into the formula: \( A = \pi \times 4 \times 2\sqrt{13} = 8\sqrt{13}\pi \mathrm{sq} \) inches.

Key Concepts

Right Circular Cone

A right circular cone is a three-dimensional geometric shape that you might find resembling party hats or ice cream cones. It features a circular base and a pointed top known as the apex. The word "right" refers to the fact that the peak is directly above the center of the base when viewed from above.

When examining a right circular cone, two important dimensions come into play:

• The base radius ( "), which is the radius of the circular base.
• The height ( "h"), which is the perpendicular distance from the apex to the center of the base.

These two parameters are essential in various calculations, such as finding the cone's volume or surface area. In this exercise, these dimensions are entirely necessary to find the slant height and ultimately, the lateral area.

Lateral Area

The lateral area of a right circular cone is a measure of the external surface area that wraps around the cone. It does not include the area of the base; rather, it pertains to the curved surface extending from the base to the apex.

The formula to calculate the lateral area is given by:

• \( A = \pi r \ell \)

where:

• \(A\) stands for the lateral area.
• \(r\) is the radius of the base.
• \(\ell\) is the slant height of the cone.

This calculation is crucial when determining how much material would be needed to create the sides of the cone if it were to be manufactured or covered with a material.

Slant Height

The slant height of a right circular cone is the diagonal length stretching from the apex down to the edge of the base. It effectively forms the hypotenuse of a right triangle, which includes the radius and the perpendicular height as its other two sides.

You can compute the slant height ("\( \ell\)") using the Pythagorean theorem. Mathematically, it is expressed as:

• \( \ell = \sqrt{r^2 + h^2} \)

where:

• \(r\) is the radius of the cone's base.
• \(h\) is the perpendicular height.

This formula arises from setting the radius and height as the two shorter sides of a right-angled triangle. The computed slant height is essential for finding the cone's lateral area as well as its complete surface area.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that helps solve problems involving right-angled triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

This relationship is expressed through the formula:

• \( c^2 = a^2 + b^2 \)

In the context of a right circular cone:

• \( c \) corresponds to the slant height (\( \ell \)).
• \( a \) refers to the height of the cone (\( h \)).
• \( b \) represents the radius of the base (\( r \)).

When applying the theorem to find the slant height of a cone, you use the radius and perpendicular height as the two known sides to calculate the unknown slant height. This practical use showcases the theorem's utility in linking different dimensions in geometric shapes.