Problem 38 - Elementary Algebra

Question

For Problems \(33-42\), match the correct formula for each statement. $$ \text { Volume of a right circular cylinder F. } A=b h $$

Step-by-Step Solution

Answer

The formula \( A = bh \) does not match the cylinder's volume formula.

Step 1: Understand the problem statement

We are given a statement about the volume of a right circular cylinder. Our task is to match this statement with a corresponding formula from a list of given choices.

Step 2: Recall the formula for the volume of a cylinder

The volume of a right circular cylinder is calculated using the formula: \[ V = \pi r^2 h \]where \( r \) is the radius of the base and \( h \) is the height of the cylinder. This involves the area of the base \( A = \pi r^2 \) and height \( h \).

Step 3: Analyze the provided formula

The formula presented in the problem is \( A = bh \). In the context of the problem, this formula is not used for the volume of a cylinder. The correct formula for a right circular cylinder involves \( \pi r^2 \), not just \( bh \).

Key Concepts

Right Circular Cylinder

A right circular cylinder is a three-dimensional geometric shape, which you can think of as a 'can' or a 'tube.' All its cross sections perpendicular to the axis of the cylinder are circles, which makes it 'circular.' It is 'right' because the axis (the imaginary line through the center of the tube) is perpendicular to the base.

• The top and bottom surfaces of this cylinder are parallel circles of equal size.
• The 'face,' or the curved surface connecting them, is a rectangle wrapped around the ends when 'unrolled.'

Understanding the attributes of a right circular cylinder is important in finding the volume, as you need both the radius of the base and the height in the calculation. The symmetry and simplicity of a right circular cylinder make it ideal for many real-world applications, such as in measuring the capacities of containers.

Cylinder Formula

The formula for the volume of a cylinder is vital, particularly in math and physics. The volume tells you how much space is inside the cylinder. The formula for finding it is: \[ V = \pi r^2 h \]Where:

• \( V \) is the volume
• \( r \) is the radius of the base
• \( h \) is the height of the cylinder

Breaking it down, the formula combines the base area, which is a circle, with the height of the cylinder. The area of a circle is given by \( \pi r^2 \), so essentially, you are stacking these circular areas atop one another up to a height of \( h \). Thus, this formula efficiently gives you the total space inside.

Problem Solving in Algebra

Solving problems involving geometric shapes often requires algebraic skills. Let's break down how to approach such problems. First, you need to understand the formulas you're working with and what each variable represents.

• Begin by identifying what you know: the given values for the dimensions of the shape.
• Next, write down the known formula needed to find the solution—here, it is \( V = \pi r^2 h \) for a cylinder's volume.
• Substitute the known values into the formula and simplify.
• Double-check calculations for errors like misplaced decimal points or incorrect operations.
• Consider the reasonableness of your answer by comparing it to the context of the problem.

Using these algebraic steps helps to systematically solve volume-related problems, providing clarity and accuracy in your results.