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Chapter 1: Problem 19

Find the volume of a cylinder with a diameter of 15 in. and a height of \(24 \mathrm{in}\).

Short Answer

Expert verified
The volume of the cylinder is 4240.5 cubic inches.

Step by step solution

01

- Identify the formula for the volume of a cylinder

The formula to find the volume of a cylinder is \[ V = \pi r^2 h \] where \( V \) is the volume, \( \pi \) is a constant (approximately 3.14), \( r \) is the radius, and \( h \) is the height.
02

- Determine the radius of the cylinder

The diameter of the cylinder is given as 15 inches. The radius is half of the diameter. Therefore, \[ r = \frac{d}{2} = \frac{15}{2} = 7.5 \text{ inches} \]
03

- Plug in the values to the volume formula

Now, we can substitute the radius \( r = 7.5 \text{ inches} \) and the height \( h = 24 \text{ inches} \) into the formula: \[ V = \pi (7.5)^2 (24) \]
04

- Calculate the volume

First, calculate the square of the radius: \[ (7.5)^2 = 56.25 \] Then, multiply by the height: \[ 56.25 \times 24 = 1350 \] Finally, multiply by \( \pi \) (approximately 3.14): \[ V = 1350 \times 3.14 = 4240.5 \text{ cubic inches} \]

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Key Concepts

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

Cylinder Volume Formula
To find the volume of a cylinder, we use a specific formula. This formula is crucial in geometry and is quite simple to remember. The formula for the volume of a cylinder is given by


substantiate substantiate substantiate Insert the radius value and height into the formula, then calculate to find the volume.
Radius Calculation
Determining your cylinder's radius is an essential step. Given the diameter, you can easily find the radius. Remember, the radius is half of the diameter.

For example:
  • A cylinder has a diameter of 15 inches.
  • The radius will be half of this diameter, which is calculated as follows:

    Understanding and calculating the radius provides the foundation for finding the cylinder's total volume.
Geometry in Algebra
Geometry often intersects with algebra, especially in problems like finding the volume of a cylinder. To blend these two fields, we need to recognize shapes, understand their properties, and apply algebraic formulas.

Here鈥檚 how these concepts combine:
  • Identify the shape (in this case, a cylinder).
  • Recall the volume formula involving algebraic terms.
  • Substitute the geometric measurements into the algebraic formula.

Both algebra and geometry contribute to a comprehensive understanding and solving of the problem effectively.
Unit Conversion
Paying attention to units is critical when calculating volume. Always ensure that measurements are in the same units.

Here's a quick guide:
  • Height and diameter/radius must be in the same unit (e.g., inches in our case).
  • Volume will be in cubic units of the chosen measurement system (e.g., cubic inches).

Consistency in units guarantees the accuracy of your final answer.

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Most popular questions from this chapter

Explain the difference between a rational number and a irrational number.

Problem: Find the volume of a cylinder with a diameter of 14 in. and a height of \(2 \mathrm{ft}\). Write the answer in cubic inches. Use \(\pi \approx 3.14\). Incorrect Answer: \(V=\pi r^{2} h\) $$ \begin{aligned} &V=(3.14)(7 \text { in. })^{2}(2 \mathrm{ft}) \\ &V=(3.14)\left(49 \text { in. }^{2}\right)(2 \mathrm{ft}) \\ &V=307.72 \text { in. }^{3} \end{aligned} $$

The regular price of an elliptical trainer is \(\$ 899.99\). It is marked down \(18 \%\). The sales tax is \(7.4 \%\). Find the final cost of the elliptical trainer including tax. Round to the nearest hundredth.

\((16 \mathrm{~mm})(2 \mathrm{~mm})\)

Find the density of a piece of gold with a mass of \(3.50 \mathrm{~g}\) and a volume of \(0.18 \mathrm{~cm}^{3}\). Round to the nearest tenth.
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