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Chapter 9: Problem 7

If the exact volume of a right circular cylinder is \(200 \pi \mathrm{cm}^{3}\) and its altitude measures \(8 \mathrm{cm},\) what is the measure of the radius of the circular base?

Short Answer

Expert verified
The radius of the circular base is 5 cm.

Step by step solution

01

Understanding the Problem

We need to find the radius of a right circular cylinder given its volume and height. The formula for the volume of a cylinder is \( V = \pi r^2 h \). We're given that \( V = 200 \pi \, \text{cm}^3 \) and \( h = 8 \, \text{cm} \).
02

Substitute Known Values

Substitute the known values into the cylinder volume formula: \( 200 \pi = \pi r^2 \times 8 \).
03

Simplify the Equation

Divide both sides of the equation by \( \pi \) to get \( 200 = r^2 \times 8 \).
04

Isolate \( r^2 \)

Divide both sides by 8 to isolate \( r^2 \): \( r^2 = \frac{200}{8} \).
05

Calculate \( r^2 \)

Calculate \( r^2 \) by dividing 200 by 8 to get \( r^2 = 25 \).
06

Find the Radius

Take the square root of both sides to find \( r \): \( r = \sqrt{25} = 5 \, \text{cm} \).

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Key Concepts

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

Understanding a Right Circular Cylinder
A right circular cylinder is a three-dimensional shape with a circular base and parallel curved sides. It resembles a can and is commonly found in many everyday objects like cans of food or beverages. It consists of:
  • Circular Base: This is what gives the cylinder its round appearance. The distance from the center to any point on the circumference is called the radius.
  • Height (or Altitude): This is the distance from the base to the top of the cylinder, measured perpendicularly.
In a right circular cylinder, the sides are perpendicular to the base, meaning the cylinder stands upright. This right angle ensures that the calculations for volume are straightforward and simple because you can use the simple geometric formulas that depend on this upright orientation. Understanding these basic characteristics of a right circular cylinder can help in solving related problems more efficiently.
The Process of Radius Calculation
Calculating the radius of a right circular cylinder can often occur when you know other measurements, such as its volume and height. Here's how this calculation typically unfolds:
  • Given Values: First, gather all provided information, such as volume and height. This exercise gives a volume of \(200 \pi \text{cm}^3\) and a height of 8 cm.
  • Use the Cylinder Volume Equation: Recall the formula for the volume of a cylinder: \[ V = \pi r^2 h \]
  • Rearrange the Formula: Substitute the known values into the formula and solve for \(r^2\):\[ 200\pi = \pi r^2 \times 8 \]
  • Isolate \(r^2\): By dividing both sides by \(\pi\) and then by the height (8), you will isolate \(r^2\):\[ r^2 = \frac{200}{8} = 25 \]
  • Determine the Radius: Finally, take the square root of both sides:\[ r = \sqrt{25} = 5 \text{cm} \]
This step-by-step approach helps in finding the radius when you're equipped with sufficient initial details such as volume and height.
Understanding the Volume Formula
In geometry, understanding the volume formula for a right circular cylinder is crucial. The formula for the volume is given by\[ V = \pi r^2 h \]This represents the space occupied within the cylinder and is measured in cubic units.
  • Circular Area Component: The term \(\pi r^2\) refers to the area of the circular base. Here, \(\pi\) is a constant approximately equal to 3.14159, and \(r\) is the radius of the base. This area is crucial because it lays the foundation for the total volume calculation.

  • Height Component: Multiplying the base area by the height (\(h\)) gives the cylinder's volume. The height stretches the base area through the third dimension, turning the circle into a full, three-dimensional object.

By breaking down the formula into understandable parts, it becomes clear how each element contributes to the volume. This breakdown helps simplify problem-solving and reinforces understanding of geometric formulas.

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Most popular questions from this chapter

For a regular square pyramid, the slant height of each lateral face has a measure equal to that of each edge of the base. If the lateral area is 200 in \(^{2}\), find the volume of the pyramid.

Use the fact that \(r^{2}+h^{2}=\ell^{2}\) in a right circular cone (Theorem 9.3.6). The teepee has a circular floor with a radius equal to \(6 \mathrm{ft}\) and a height of \(15 \mathrm{ft}\). Find the volume of the enclosure. (figure cannot copy)

Find the total area (surface area) of a regular dodecahedron (12 faces) if the area of each face is \(6.4 \mathrm{cm}^{2}\)

Kianna's aquarium is "box-shaped" with dimensions of \(2 \mathrm{ft}\) by \(1 \mathrm{ft}\) by 8 in. If \(1 \mathrm{ft}^{3}\) corresponds to 7.5 gal of water, what is the water capacity of her aquarium in gallons?

Given that \(100 \mathrm{cm}=1 \mathrm{m},\) find the number of cubic centimeters in 1 cubic meter.
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