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Chapter 2: Problem 40

To prepare for Exercises 43 and \(44,\) use the volume formulas below to solve Exercises \(39-42 .\) Remember, volume is measured in cubic units. The battery below is in the shape of a cylinder and has an exact volume of \(825 \pi\) cubic millimeters. Find its height. (GRAPH NOT COPY)

Short Answer

Expert verified
Use the formula \( h = \frac{825}{r^2} \) to find height.

Step by step solution

01

Recall the Volume Formula for a Cylinder

The volume of a cylinder is given by the formula \( V = \pi r^2 h \), where \( V \) is the volume, \( r \) is the radius of the base, and \( h \) is the height of the cylinder.
02

Identify Given Values

We are given that the volume \( V = 825\pi \). Let us assume we know that the radius \( r \) of the cylinder, as it seems not directly provided. For example, if the radius \( r \) is a known value, we will use it in the next step.
03

Substitute Known Values into the Formula

Assume \( r \) is provided to us. Substitute \( V = 825\pi \) and \( r \) into the formula: \( 825\pi = \pi r^2 h \).
04

Solve for Height (\( h \))

Remove \( \pi \) from both sides of the equation, \( 825 = r^2 h \). Solve for \( h \) using the formula: \( h = \frac{825}{r^2} \).
05

Complete the Solution

Once the radius \( r \) is provided, plug it into the equation \( h = \frac{825}{r^2} \). For any specific numerical value of \( r \), compute the resulting height.

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Key Concepts

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

Cylinder Volume Formula
Understanding the concept of the cylinder volume formula is critical to solving problems involving cylindrical shapes. The formula is:\[ V = \pi r^2 h \]where:
  • \( V \) is the volume of the cylinder.
  • \( \pi \approx 3.14159 \) is a mathematical constant.
  • \( r \) is the radius of the cylinder's circular base.
  • \( h \) is the height of the cylinder.
This formula helps us understand how changes in the radius or height affect the overall volume of the cylinder. In practical terms, double the radius or the height, and the volume increases significantly. Keeping these relationships in mind makes it easier to manipulate and solve for unknown quantities, like height or radius, given a specific volume.
Solving for Height
Finding the height when you know the volume requires rearranging the cylinder volume formula. If we have the formula:\[ V = \pi r^2 h \]and need to solve for \( h \), it's crucial to isolate \( h \) on one side of the equation. By doing some simple algebra, we can rearrange it as follows:
  • First, remove \( \pi \) by dividing both sides by \( \pi \): \[ \\frac{V}{\pi} = r^2 h \]

  • Next, divide both sides by \( r^2 \) to solve for \( h \): \[ \h = \frac{V}{\pi r^2} \]
This is the equation you'll use to find the cylinder's height once you know its volume and radius. Remember, it's important to ensure your radius and volume measurements are in the same unit system for accuracy.
Volume Measurement
Understanding the units of volume measurement is key when dealing with cylinders and other 3D shapes. Volume is always expressed in cubic units, because it represents not just length or area, but a full three-dimensional space.
  • This might be cubic millimeters (mm鲁), cubic centimeters (cm鲁), or even cubic meters (m鲁), depending on the context of the problem.
It is important to always keep the unit consistent throughout the calculations to avoid mistakes in your final result.
For example, if a cylinder's dimensions are measured in millimeters, then the volume should also be calculated in cubic millimeters. Paying attention to units helps ensure your solution remains accurate and relevant to real-world applications.
Algebraic Equations
Algebraic equations are a powerful tool for solving problems involving unknowns, such as finding the height in a volume problem. When solving equations like:\[ 825 = r^2 h \],it鈥檚 important to understand how to manipulate the equation to isolate the desired variable, in this case, \( h \). This often involves:
  • Using division or multiplication across the equation to isolate the unknown.
  • Being methodical: apply the same operation to both sides of the equation to maintain equality.

  • Checking your solution by substituting back into the original equation when possible.
By mastering these techniques, you can confidently solve a wide range of problems not just in geometry, but in other fields that require similar mathematical reasoning.

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