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Chapter 12: Problem 17

A cylinder has a diameter of 32 feet and a height of 20 feet. a. Find the lateral area of the cylinder. b. Find the surface area of the cylinder.

Short Answer

Expert verified
a. 640蟺 sq ft; b. 1152蟺 sq ft

Step by step solution

01

Find the radius of the cylinder

The radius is half of the diameter. The diameter is given as 32 feet, so divide by 2 to get the radius:\[ \text{Radius} = \frac{32}{2} = 16 \text{ feet} \]
02

Calculate the lateral area of the cylinder

The lateral area of a cylinder is given by the formula \( 2 \pi r h \), where \( r \) is the radius and \( h \) is the height. Use \( r = 16 \) feet and \( h = 20 \) feet to substitute into the formula:\[ \text{Lateral Area} = 2 \pi (16)(20) = 640 \pi \text{ square feet} \]
03

Calculate the areas of the cylinder's top and bottom

Each end of the cylinder is a circle, and the area of a circle is \( \pi r^2 \). Calculate the area of one end using the radius from Step 1:\[ \text{Area of one end} = \pi (16)^2 = 256\pi \text{ square feet} \]Since there are two ends, multiply by 2:\[ \text{Total area of top and bottom} = 2 \times 256\pi = 512\pi \text{ square feet} \]
04

Calculate the total surface area of the cylinder

The total surface area of a cylinder is the sum of its lateral area and the areas of its two ends. Add the lateral area from Step 2 and the areas of the ends from Step 3:\[ \text{Surface Area} = 640\pi + 512\pi = 1152\pi \text{ square feet} \]

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Key Concepts

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

Lateral Area
The lateral area of a cylinder is the surface area that wraps around the cylindrical shape, excluding the top and bottom ends. Imagine the label around a can. That's your lateral area! To find this, we use a simple formula:
  • Start with the formula: \( 2 \pi r h \) \( (r) \) is the radius, and \( (h) \) is the height of the cylinder.
  • Here, the symbol \( \pi \) represents the mathematical constant Pi (approximately 3.14159).
For our exercise, we calculated:
  • Radius \( (r) \) = 16 feet, Height \( (h) \) = 20 feet
  • Plug these values into the formula: \( 2 \pi (16)(20) \)
  • The lateral area results in \( 640\pi \) square feet.
This area is important if you're determining materials needed to wrap something around the cylinder.
Cylinder Height
The height of a cylinder is the distance between its two circular bases. Think of it as how tall the cylinder stands. It's a straightforward concept but crucial for calculating both the lateral and the total surface area of the cylinder.
  • In this problem, the height \((h)\) is given as 20 feet.
  • It's used in the calculation for the lateral area: \( 2 \pi r h \).
Remember:
  • The height doesn't affect the size of the top and bottom circles, but it greatly influences the lateral area.
  • A taller cylinder will have more surface to "wrap," increasing its lateral area.
Understanding height helps in various real-world scenarios, like optimizing storage space or designing labels.
Cylinder Radius
The radius of a cylinder is a key measurement, as it is half the diameter of the circular base. For our problem:
  • Given a diameter of 32 feet, the radius \((r)\) is half of this: \( \frac{32}{2} = 16 \) feet.
  • This radius is crucial for multiple calculations, like finding the lateral area and the areas of the top and bottom circles.
For calculation purposes:
  • The lateral area depends on the radius: \( 2 \pi r h \)
  • The area of each circle at the ends (top and bottom) is \( \pi r^2 \)
In our example, the radius was used to calculate:
  • Lateral Area: \( 2 \pi (16)(20) = 640\pi \)
  • Area of each circle: \( \pi (16)^2 = 256\pi \)
Understanding the radius is essential when dealing with any sort of round objects or shapes.

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Most popular questions from this chapter

Science The diameter of Earth is about 7900 miles. a. Find the surface area of Earth to the nearest hundred square miles. b. Find the volume of Earth to the nearest hundred cubic miles. c. Most of Earth's atmosphere is less than 50 miles above the surface. Find the volume of Earth's atmosphere to the nearest hundred cubic miles.

Draw a cone in which the altitude is not perpendicular to the base at its center. What is the name of this solid?

Draw a rectangular prism that is 4 centimeters by 5 centimeters by 8 centimeters. Find the surface area of the prism.

Find the volume of a sphere with a radius of 10 inches. Round to the nearest hundredth.

Compare and contrast circles and spheres.
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