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Chapter 9: Problem 289

Find a the volume and b the surface area of the cylinder with the given radius and height. Round answers to the nearest hundredth. radius 1.5 meters, height 4.2 meters

Short Answer

Expert verified
Volume: 29.68 cubic meters. Surface Area: 53.69 square meters.

Step by step solution

01

Identify the formulas

To find the volume and the surface area of a cylinder, use the following formulas: Volume: \( V = \pi r^2 h \) Surface Area: \[ SA = 2\pi rh + 2\pi r^2 \]
02

Substitute the values into the volume formula

Given radius \( r = 1.5 \) meters and height \( h = 4.2 \) meters, substitute these values into the volume formula: \[ V = \pi (1.5)^2 (4.2) \]
03

Calculate the volume

First, calculate \( 1.5^2 \): \[ 1.5^2 = 2.25 \] Then multiply by the height and \ \pi \: \[ V = \pi \times 2.25 \times 4.2 = \pi \times 9.45 \] Using \ \pi \approx 3.14 \: \[ V \approx 3.14 \times 9.45 = 29.68 \] So, the volume is approximately 29.68 cubic meters.
04

Substitute the values into the surface area formula

Now use the surface area formula: \[ SA = 2\pi rh + 2\pi r^2 \] Substitute the values: \[ SA = 2\pi (1.5)(4.2) + 2\pi (1.5)^2 \]
05

Calculate the surface area

First calculate \( 2\pi (1.5)(4.2) \): \[ 2\pi \times 6.3 \approx 2 \times 3.14 \times 6.3 = 39.564 \] Then, calculate \( 2\pi (1.5)^2 \): \[ 2\pi \times 2.25 \approx 2 \times 3.14 \times 2.25 = 14.13 \] Add the two results: \[ SA = 39.564 + 14.13 = 53.694 \] So, the surface area is approximately 53.69 square meters.

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Key Concepts

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

cylinder volume
The volume of a cylinder measures the amount of space inside it. To find the volume, we use the formula: \[ V = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height. The formula breaks down into several steps:
  • First, square the radius value, which means to multiply it by itself.
  • Next, multiply this squared value by the height of the cylinder.
  • Finally, multiply by \( \pi \) (approximately 3.14) to get the volume.
Applying this to our problem, with \( r = 1.5 \) meters and \( h = 4.2 \) meters, we find: \[ V = \pi (1.5)^2 (4.2) \] First, calculate \( (1.5)^2 = 2.25 \). Then multiply by 4.2, which gives us 9.45. Finally, multiply by \( \pi \): \[ V = \pi \times 9.45 \approx 29.68 \] Thus, the volume is about 29.68 cubic meters.
cylinder surface area
The surface area of a cylinder includes both the area of its side and the areas of its two bases. To find it, we use: \[ SA = 2\bpi rh + 2\pi r^2 \] where \( r \) is the radius and \( h \) is the height. This formula breaks down into:
  • Calculating the lateral surface area: \( 2\pi rh \)
  • And adding the area of the two bases: \( 2\pi r^2 \)
First, calculate for the given radius \( r = 1.5 \) meters and height \( h = 4.2 \) meters: \[ SA = 2\pi (1.5) (4.2) + 2\pi (1.5)^2 \] Breakdown:
  • First part: \( 2\pi (1.5)(4.2) = 2\pi \times 6.3 \)
  • Second part: \( 2\pi (1.5)^2 = 2\pi \times 2.25 \)
So, the calculations will be: \[ 2\pi \times 6.3 = 39.564 \] \[ 2\pi \times 2.25 = 14.13 \] Adding these together gives: \[ SA = 39.564 + 14.13 = 53.69 \] Therefore, the surface area is approximately 53.69 square meters.
geometry formulas
Understanding the formulas for different shapes is crucial in geometry. Cylinders have two main formulas: one for volume and one for surface area.
  • Volume: \( V = \pi r^2 h \)
  • Surface Area: \( SA = 2\pi rh + 2\pi r^2 \)
These formulas rely on knowing simple geometric properties like radius and height. Master these equations by practicing various problems and learning how to apply known values into the formulas. Geometry often combines multiple concepts and helps visualize mathematical relationships.
mathematical calculations
Performing mathematical calculations correctly is essential when solving problems. Steps to ensure accuracy include:
  • Breaking down the problem into smaller steps.
  • Double-checking each calculation.
  • Using approximations of constants like \( \pi \) (approximately 3.14).
In geometrical problems, calculations often involve:
  • Squaring numbers (multiplying a number by itself).
  • Multiplying several values together.
  • Managing units, like meters for length or square meters for area.
For our cylinder example, we accurate calculation allows us to determine the volume and surface area precisely. Always double-check each step and use appropriate tools, like a calculator, if needed. This will help avoid errors and achieve correct results.

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