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Chapter 8: Problem 6

find the surface area of each cylinder. Round answer to tenth of a square unit. \(r=3.4 \mathrm{m}, h=10.5 \mathrm{m}\)

Short Answer

Expert verified
297.1 square meters

Step by step solution

01

- Identify the surface area formula for a cylinder

The surface area (SA) of a cylinder can be found using the formula: \[ SA = 2\pi r(h + r) \] where \( r \) is the radius and \( h \) is the height.
02

- Plug in the given values

Given \( r = 3.4 \, \text{m} \) and \( h = 10.5 \, \text{m} \), substitute these values into the formula: \[ SA = 2\pi (3.4)(10.5 + 3.4) \]
03

- Calculate the expression inside the parentheses

Add the radius and height: \[ 10.5 + 3.4 = 13.9 \, \text{m} \]
04

- Substitute back and multiply

Now substitute \( 13.9 \) back into the formula: \[ SA = 2\pi (3.4)(13.9) \] Multiply the terms to get: \[ SA = 2 \times 3.4 \times 13.9 \times \pi \]
05

- Calculate the exact surface area

Calculate the product: \[ 2 \times 3.4 \times 13.9 = 94.52 \] Thus, \[ SA = 94.52 \pi \] or \[ SA \approx 94.52 \times 3.14159 = 297.07 \, \text{square meters} \]
06

- Round to the nearest tenth

Round the result to the nearest tenth: \( 297.07 \approx 297.1 \, \text{square meters} \)

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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, we use a specific formula. This formula is derived from the geometry of the cylinder, which consists of two circles and a rectangle. The surface area (SA) formula for a cylinder is: \( SA = 2\pi r(h + r) \).
Here:
  • \( r \) is the radius of the cylinder's base.
  • \( h \) is the height of the cylinder.
By using this formula, we can calculate the total surface area, which includes both the lateral surface (side) and the area of the two circular bases.
Cylinder Geometry
Understanding cylinder geometry will simplify our calculation. A cylinder is a 3-dimensional shape with two parallel circular bases connected by a curved surface.
The main components of a cylinder are:
  • The radius (\( r \)): The distance from the center of the base to the perimeter.
  • The height (\( h \)): The distance between the two bases, measured perpendicular to them.
To wrap your head around it, imagine a can. The circles at the top and bottom are the bases, and the height is how tall the can stands.
Step-by-Step Math Solutions
Breaking down the computation into steps helps ensure accuracy and understanding.
Here's a step-by-step approach to finding the surface area of a cylinder:
  • Step 1: Begin by establishing the formula (\( SA = 2\pi r(h + r) \)).
  • Step 2: Substitute the given values (\( r \) and \( h \)) into the formula.
  • Step 3: Calculate the sum inside the parenthesis (\( h + r \)).
  • Step 4: Use this result to update the formula (\( 2\pi r \times (h + r) \)).
  • Step 5: Multiply all terms to get the surface area in terms of \(\pi\).
  • Step 6: Multiply by \(\pi\) (\(3.14159\)) for a final numerical answer.
  • Step 7: Round to the nearest tenth for precision.
Following these steps ensures you don鈥檛 miss any critical calculations.
Radius and Height Substitution
Substituting given values into the formula is a straightforward but crucial step in solving cylinder surface area problems. Let's go through it with our given values:
  • Radius (r): 3.4 meters
  • Height (h): 10.5 meters
Plug these values into the formula (\( SA = 2\pi r(h + r) \)) like this:
\( SA = 2\pi(3.4)(10.5 + 3.4) \).
First, add the height and radius:
\( 10.5 + 3.4 = 13.9 \).
Then multiply: \( 2 \times 3.4 \times 13.9 \times \pi \).
This simplifies our formula and ensures accurate results.

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Most popular questions from this chapter

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