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Chapter 9: Problem 26

Use the fact that \(r^{2}+h^{2}=\ell^{2}\) in a right circular cone (Theorem 9.3.6). A triangle has sides that measure \(15 \mathrm{cm}, 20 \mathrm{cm},\) and \(25 \mathrm{cm} .\) Find the exact volume of the solid of revolution formed when the triangle is revolved about the side of length \(20 \mathrm{cm}\)

Short Answer

Expert verified
The exact volume of the cone is \( 1500\pi \) cm鲁.

Step by step solution

01

Identify the Right Triangle

The given triangle with sides 15 cm, 20 cm, and 25 cm is a right triangle. We identify the hypotenuse, which is the longest side, as 25 cm. The side of length 20 cm will act as the diameter of the base when the triangle is revolved to form a cone.
02

Determine the Cone's Dimensions

Given the rotation about the 20 cm side, this will be the height (h) of the cone, and the other leg, 15 cm, will be the radius (r) of the cone's base since a circle is formed with this side as the radius when the triangle is revolved.
03

Start Formula for Volume of Cone

The volume of a cone is given by the formula: \( V = \frac{1}{3} \pi r^2 h \). Here, \( r = 15 \) cm and \( h = 20 \) cm.
04

Calculate Volume

Plug the radius and height into the formula: \( V = \frac{1}{3} \pi (15)^2 (20) \). Calculate to get \( V = \frac{1}{3} \pi (225)(20) \).
05

Simplify Calculation

Simplify the expression to find the volume: \( V = \frac{1}{3} \times 4500 \pi = 1500\pi \).
06

State the Exact Volume

The exact volume of the solid of revolution (cone) formed is \( 1500\pi \) cubic centimeters.

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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 three-dimensional geometric shape that features a circular base and a pointed top called the apex. When visualizing a cone, think of it as a classic ice cream cone. The term "right" refers to the fact that the altitude or height of the cone is perpendicular to its base. This distinguishes it from an oblique cone, where the apex is not aligned above the center of the base.

In a right circular cone:
  • The base is circular, defined by a radius.
  • The height (h) is a straight line from the base to the apex, perpendicular to the base.
  • The slant height (\(\ell\)) connects the apex to any point on the circumference of the base.
Understanding these elements is crucial when calculating or visualizing properties of the cone, such as its volume, surface area, or when involved in transformations such as revolutions.
Solid of Revolution
A solid of revolution is formed when a shape rotates around an axis, creating a symmetrical three-dimensional object. This concept allows us to visualize shapes in new dimensions and is especially useful in geometry for volume calculations.

To better understand:
  • The process involves rotating a two-dimensional figure, like a triangle or rectangle, around a given axis.
  • This rotation sweeps out a region in space, effectively "filling" the area into a solid form.
  • In the exercise, the triangle revolved around one of its sides (20 cm), creating a cone.
Understanding the principles of solids of revolution aids in visualizing how basic shapes can transform into more complex objects, facilitating their study and manipulation.
Volume Calculation
Determining the volume of a right circular cone involves a straightforward formula. The volume (\(V\)) is calculated using:\[V = \frac{1}{3} \pi r^2 h\]where
  • \(r\) is the radius of the base.
  • \(h\) is the height of the cone.
To apply this formula in the given problem:
  • The base radius (\(r\)) is 15 cm.
  • The height (\(h\)) is 20 cm.
Plug these values into the formula:\[V = \frac{1}{3} \pi (15)^2 (20)\]First, calculate the square of the radius:\[(15)^2 = 225\]Then, multiply by the height:\[225 \times 20 = 4500\]Finally, apply the cone volume formula:\[V = \frac{1}{3} \times 4500 \pi = 1500\pi\]Thus, the volume of the cone is \(1500\pi\) cubic centimeters.
Right Triangle Properties
Right triangles are fundamental in geometry because their properties simplify calculations and proofs. In mathematics, these triangles have an angle of 90 degrees, forming two perpendicular sides, typically referred to as the legs, and the longest side called the hypotenuse.

Key properties include:
  • The Pythagorean Theorem, \(a^2 + b^2 = c^2\), which relates the lengths of the sides. Here, \(a\) and \(b\) are the legs, and \(c\) is the hypotenuse.
  • The hypotenuse is always the longest side.
  • Right triangles often serve as building blocks in solving problems related to geometry, trigonometry, and construction.
In the given exercise, recognizing the right triangle allowed us to correctly assign the dimensions for forming a cone. The triangle with sides 15 cm, 20 cm, and 25 cm was confirmed as a right triangle since \(15^2 + 20^2 = 25^2\).

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Most popular questions from this chapter

Where \(e_{1}\) and \(e_{2}\) are the lengths of two corresponding edges (or altitudes) of similar prisms or pyramids, the ratio of their volumes is \(\frac{V_{1}}{V_{2}}=\left(\frac{e_{1}}{e_{2}}\right)^{3} .\) Write a ratio to compare volumes for two similar regular square pyramids in which \(e_{1}=4\) in. and \(e_{2}=2\) in.

Use the calculator value of \(\pi\). For a sphere whose radius has length \(7 \mathrm{cm},\) find the approximate a) surface area. b) volume.

Sketch the solid that results when the given circle of radius 1 is revolved about the horizontal line that lies 1 unit below the center of that circle. (Diagram can't copy)

Make drawings as needed. How many common tangent planes do two externally tangent spheres have?

A raised flower bed is 2 ft high by 12 ft wide by \(15 \mathrm{ft}\) long. The mulch, soil, and peat mixture used to fill the raised bed costs \(\$ 9.60\) per cubic yard. What is the total cost of the ingredients used to fill the raised garden?
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