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Chapter 11: Problem 9

Find the volume of the cone. A cone has a diameter of 11.5 inches and a height of 15.2 inches.

Short Answer

Expert verified
The volume of the cone is approximately 502.65 cubic inches.

Step by step solution

01

Calculate Radius

The radius of a cone is half the diameter, thus: \[r = \frac{d}{2}\], where d is the diameter. Substituting 11.5 inches for the diameter, the radius \(r = \frac{11.5}{2} = 5.75\) inches
02

Substitute into the Volume Formula

Next, substitute the radius of 5.75 inches and height of 15.2 inches to the formula \[V = \frac{1}{3} \pi r^2 h\], which yields that \[V = \frac{1}{3} \pi (5.75)^2 \times 15.2\]
03

Calculate the Volume

Perform the calculations within the formula to determine the volume of the cone. The volume is approximately = 502.65 cubic inches.

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Key Concepts

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

Geometry Formulas
Understanding geometry formulas is essential in solving problems related to shapes, including cones. For a cone, we focus on the volume formula which tells us how much space is inside the cone.
The formula for the volume of a cone is \[ V = \frac{1}{3} \pi r^2 h \]where:
  • \( V \) is the volume of the cone
  • \( r \) is the radius of the base of the cone
  • \( h \) is the height of the cone
  • \( \pi \) is a mathematical constant approximately equal to 3.14159
This formula is crucial because it helps convert the physical dimensions of a cone into a numerical value representing its volume, offering insight into its size in three-dimensional space. Understanding each part of the formula allows one to accurately calculate volumes with ease.
Cone Measurement
Measuring a cone involves determining its key dimensions: radius and height. These measurements are critical because they are inputs for the volume formula.
The radius of a cone is half the diameter of its base. In mathematical terms, given the diameter \( d \), the radius \( r \) is calculated as \( r = \frac{d}{2} \). In our example, with a diameter of 11.5 inches, the radius is:
  • \( r = \frac{11.5}{2} = 5.75 \) inches
The height \( h \) is the perpendicular distance from the base to the tip of the cone. For our cone, the height is given as 15.2 inches.
Measuring these dimensions accurately is essential because even small errors can lead to incorrect volume calculations.
Calculation of Volume
To calculate the volume of a cone, we use the formula for volume \( V = \frac{1}{3} \pi r^2 h \). Once you have the radius and height measurements, substitute them into the formula.
From our previous calculations:
  • Radius, \( r = 5.75 \) inches
  • Height, \( h = 15.2 \) inches
Substitute these into the formula:\[V = \frac{1}{3} \pi (5.75)^2 \times 15.2\]Calculate \( (5.75)^2 = 33.0625 \), then multiply by 15.2 to get the product 502.65, and finally multiply by \( \frac{1}{3} \pi \). This will give the approximate volume of the cone as 502.65 cubic inches.
This process illustrates how geometry formulas and measurement practice work together to determine the volume of a cone accurately.

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Most popular questions from this chapter

In Exercises 19 and 20, \(\square\)nd the volume of the sphere with the given surface area. (See Example 4 .) Surface area \(=16 \pi \mathrm{ft}^{2}\)

A frustum of a cone is the part of the cone that lies between the base and a plane parallel to the base, as shown. Write a formula for the volume of the frustum of a cone in terms of, b, and h. (Hint: Consider the "missing" top of the cone and use similar triangles.)

In Exercises 11 and \(12,\) make a sketch of the solid and Ind its volume. Round your answer to the nearest hundredth. A prism has a height of 11.2 centimeters and an equilateral triangle for a base, where each base edge is 8 centimeters.

In Exercises 11 and \(12,\) make a sketch of the solid and Ind its volume. Round your answer to the nearest hundredth. A pentagonal prism has a height of 9 feet and each base edge is 3 feet.

Solve the triangle. Round decimal answers to the nearest tenth. $$\mathrm{B}=102^{\circ}, \mathrm{C}=43^{\circ}, \mathrm{b}=21$$
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