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Chapter 9: Problem 30

Use the fact that \(r^{2}+h^{2}=\ell^{2}\) in a right circular cone (Theorem 9.3.6). A right circular cone has a slant height of \(12 \mathrm{ft}\) and a lateral area of \(96 \pi \mathrm{ft}^{2}\). Find its volume. (figure cannot copy)

Short Answer

Expert verified
The cone's volume is \( \frac{256}{3} \pi \sqrt{5} \approx 119.12 \pi \) \(\text{ft}^3\).

Step by step solution

01

Understand the Given Information

We have a right circular cone with a slant height \( \ell = 12 \) ft and a lateral area \( A = 96 \pi \) ft虏. We are tasked with finding the cone's volume. We will use the relationship \( r^2 + h^2 = \ell^2 \) and the formula for lateral surface area \( A = \pi r \ell \).
02

Use Lateral Area to Find Radius

The formula for lateral area of a cone is \( A = \pi r \ell \). We substitute the known values into this formula: \( 96\pi = \pi r \cdot 12 \). Simplifying gives \( r = 8 \) ft.
03

Calculate Height Using Pythagorean Theorem

With \( r = 8 \) and \( \ell = 12 \), use the relation \( r^2 + h^2 = \ell^2 \) to find the height. Substitute the values: \( 8^2 + h^2 = 12^2 \), giving \( 64 + h^2 = 144 \). Solve for \( h^2 \) to get \( h^2 = 80 \), thus \( h = \sqrt{80} = 4\sqrt{5} \) ft.
04

Calculate Volume of the Cone

The volume of a cone is given by \( V = \frac{1}{3} \pi r^2 h \). Substitute \( r = 8 \) and \( h = 4\sqrt{5} \) into this formula, we have \( V = \frac{1}{3} \pi (8)^2 (4\sqrt{5}) \). Simplify to \( V = \frac{1}{3} \pi \times 64 \times 4\sqrt{5} = \frac{1}{3} \times 256\pi \sqrt{5} \).
05

Calculate Final Volume

Multiply through to find the final volume. \( V = \frac{256}{3} \pi \sqrt{5} \approx 119.12 \pi \) cubic \(\text{ft}^3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Right Circular Cone
A right circular cone is a 3-dimensional geometric shape that resembles an ice cream cone. It has a circular base and a pointed top called the apex. The height (\( h \)) is the perpendicular distance from the base to the apex. The slant height (\( \ell \)) is the hypotenuse of the triangle formed by the radius (\( r \)), height, and slant height.
Right circular cones are common in both math problems and real-life scenarios. Understanding their properties helps in solving problems related to volume and surface area calculations.
  • Base: Circular with radius (\( r \)).
  • Height: Perpendicular distance from base to apex.
  • Slant height: Hypotenuse of the right triangle.
Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in geometry, particularly useful when working with right triangles. In the context of a right circular cone, it relates the radius (\( r \)), height (\( h \)), and slant height (\( \ell \)).
For a cone, the equation \( r^2 + h^2 = \ell^2 \) forms the basis for finding unknown dimensions if two others are known. It allows us to calculate the cone's height when the radius and slant height are given.
  • Set up the equation: Given (\( r \)), solve for height (\( h \)) using \( r^2 + h^2 = \ell^2 \)
  • Calculate: Rearrange the equation and solve for \( h \)
This theorem is essential for ensuring accuracy in dimensional calculations of the cone.
Volume Calculation
Finding the volume of a right circular cone involves understanding and applying the correct formula, which is\[V = \frac{1}{3} \pi r^2 h\]The formula multiplies the area of the cone's base (\( \pi r^2 \)) by the height, and this product is then divided by three. This accounts for the cone's tapering shape.
  • Determine the radius (\( r \)) and height (\( h \)).
  • Calculate the base area: \( \pi r^2 \)
  • Use the formula: Substitute \( r \) and \( h \) into the volume formula.
By using this method, one can compute the volume of any right circular cone efficiently.
Lateral Surface Area
The lateral surface area of a cone refers to the area of its surface excluding the base. For a right circular cone, this calculation is crucial for understanding the cone's outer dimensions.
The formula used is:\[A = \pi r \ell\]This formula allows us to determine the area by multiplying the base's radius by the slant height and \( \pi \) . It鈥檚 especially useful for practical applications, like determining how much material is needed to cover the cone.
  • Identify the radius (\( r \)) and slant height (\( \ell \)).
  • Plug into the formula: Compute \( A \) using \( \pi r \ell \).
Understanding this concept helps in solving problems that deal with material requirements or surface coatings for conical shapes.

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Most popular questions from this chapter

A sphere has a volume equal to \(\frac{99}{7}\) in \(^{3}\). Determine the length of the radius of the sphere. (Let \(\pi \approx \frac{22}{7}\).)

Does a right circular cone such as a wizard's cap have a. symmetry with respect to at least one plane? b. symmetry with respect to at least one line? c. symmetry with respect to a point?

Use the formula from Exercise 41 Similar triangles were used to find \(h\) and \(H\) A lawn roller in the shape of a right circular cylinder has a radius of 18 in. and a length (height) of \(4 \mathrm{ft}\). Find the area rolled during one complete revolution of the roller. Use the calculator value of \(\pi,\) and give the answer to the nearest square foot. (figure cannot copy)

Use the information from Exercise 39 to find the ratio of volumes \(\frac{V_{1}}{V_{2}}\) for two cubes in which \(e_{1}=2 \mathrm{cm}\) and \(e_{2}=6 \mathrm{cm}\).

Suppose that the base of the hexagonal pyramid in Exercise 6 has an area of \(41.6 \mathrm{cm}^{2}\) and that the altitude of the pyramid measures \(3.7 \mathrm{cm} .\) Find the volume of the hexagonal pyramid.
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