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Chapter 8: Problem 4

find the surface area of each cylinder. Round answer to tenth of a square unit. $$r=9 \mathrm{cm}, h=15 \mathrm{cm}$$

Short Answer

Expert verified
1356.5 square cm

Step by step solution

01

- Understand the formula

The surface area of a cylinder is given by the formula: \[A = 2\pi r(h + r)\]where \(r\) is the radius and \(h\) is the height.
02

- Substitute the given values

Substitute the given radius \(r = 9\) cm and height \(h = 15\) cm into the formula: \[A = 2\pi(9)(15 + 9)\]
03

- Simplify inside the parentheses

First, add the values inside the parentheses: \[15 + 9 = 24\]So the formula becomes: \[A = 2\pi(9)(24)\]
04

- Multiply the values together

Now, multiply the numbers together: \[2 \times 9 \times 24 = 432\]So the formula becomes: \[A = 432\pi\]
05

- Calculate the exact surface area

Use the value of \(\pi \approx 3.14\) to find the approximate surface area: \[A = 432 \times 3.14\]
06

- Perform the multiplication

Multiply 432 by 3.14 to get the approximate surface area: \[A \approx 1356.48\]
07

- Round to the nearest tenth

Round 1356.48 to the nearest tenth to get: \[A \approx 1356.5\]The surface area is approximately 1356.5 square cm.

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Key Concepts

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

Surface Area Formula
To find the surface area of a cylinder, you need to understand the formula: The surface area, denoted as A, is calculated with the formula \[A = 2\pi r(h + r)\]Here, \(r\) represents the radius of the circular base, while \(h\) stands for the height of the cylinder. The formula combines the areas of the two circular bases and the rectangular side surface wrapped around the cylinder.
By using this formula, we can efficiently determine the total area covering the surface of any cylinder when given the radius and height.
Geometry
Geometry plays a crucial role in understanding cylinders. A cylinder has two main properties:
  • It has two circular bases that are parallel to each other.
  • A side rectangle that wraps around the bases to form a curved surface.
The circular bases have a defined radius \(r\), and the height \(h\) is the distance between the two bases.
These foundational elements help us use geometric formulas efficiently to solve problems related to surface areas of cylinders.
Cylinder Properties
A cylinder, a 3D solid shape, is defined by the radius of its circular base and its height.
  • The radius (\(r\)) is the distance from the center to the edge of the base.
  • The height (\(h\)) stretches from one base to the other, measured perpendicularly to the bases.
Understanding these properties makes it easier to apply surface area calculations.
Also, recognize that a cylinder's surface area includes both the curved surface and the total area of the two circular bases.
Mathematical Calculations
Executing the surface area calculation involves multiple steps:
  1. First, start with the given formula: \[A = 2\pi r(h + r)\]
  2. Next, substitute the given values: \(r = 9 \) cm and \(h = 15 \) cm.
  3. Simplify calculations inside parentheses: \[15 + 9 = 24\]
  4. Multiply: \[2 \times 9 \times 24 = 432\]
  5. Finally, include \(\pi\) by approximating its value: \[A = 432 \pi\]
  6. For a more precise result, use \(\pi \approx 3.14\): \[A \approx 432 \times 3.14 = 1356.48\]
  7. Round the final result to the nearest tenth: 1356.5 square cm.
Breaking down each step clarifies how each calculation builds on the previous one.
Rounding Numbers
Rounding numbers is essential for simplifying results. When rounding to the nearest tenth:
  • Look at the digit in the hundredths place.
  • If it's 5 or higher, round up the digit in the tenths place.
  • If it's lower than 5, keep the tenths place digit the same.
In the example, 1356.48 has 8 in the hundredths place, thus we round up, making it 1356.5 square cm.
Rounding helps present easier and cleaner results, especially when dealing with real-world applications.

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Most popular questions from this chapter

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