Problem 49 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Solve for \(h: V=\pi r^{2} h\)

Step-by-Step Solution

Answer

The formula for \( h \) is \( h = \frac{V}{\pi r^2} \).

Step 1: Understand the Equation

The given equation is a volume formula for a cylinder: \( V = \pi r^2 h \). In this equation, \( V \) is the volume, \( r \) is the radius of the base, and \( h \) is the height, which is the variable we need to solve for.

Step 2: Isolate the Variable

To solve for \( h \), we need to isolate it on one side of the equation. Start by dividing both sides of the equation by \( \pi r^2 \) to get:\[h = \frac{V}{\pi r^2}\]

Step 3: Rearrange and Write the Solution

After isolating \( h \), the equation becomes \( h = \frac{V}{\pi r^2} \). This is the expression for \( h \) in terms of \( V \) and \( r \).

Step 4: Verify the Rearranged Formula

To verify, substitute \( h \) back into the original equation and check that both sides are equal, indicating that the rearrangement was done correctly.

Key Concepts

Solving Equations for a Variable

When faced with a mathematical equation, solving for a variable means isolating it on one side of the equation. In the context of the given problem, our goal is to find the expression for the variable

• Identify the variable you need to solve for: In this exercise, it was the height, represented by \( h \).
• Rearrange the equation to make the variable the subject: This involves making sure that only the variable appears on one side of the equation.
• Use algebraic operations such as addition, subtraction, multiplication, and division to isolate the variable.

Specifically for this problem, we divided both sides by \( \pi r^2 \) to isolate \( h \), resulting in the formula \( h = \frac{V}{\pi r^2} \). These steps are crucial as they form the backbone of solving any algebraic equations.

Surface Area and Volume

In geometry, understanding the concepts of surface area and volume allows us to describe the size of a three-dimensional shape. Volume refers to the amount of space occupied by an object, while surface area is related to the sum of areas covering the external parts of the object.

For a cylinder:

• The volume is calculated using the formula \( V = \pi r^2 h \), where \( V \) stands for volume, \( r \) the radius, and \( h \) the height.
• The surface area involves calculating both the lateral (side) surface and the top/bottom surfaces, but that wasn't part of our original equation focus.

Having a grasp of these formulas helps us not only solve mathematical problems but also understand real-world scenarios where these calculations might be applied, such as determining how much space is inside a can or how much material is needed to cover its outside.

Geometry in Algebra

Geometry and algebra intersect beautifully when equations are used to represent geometric shapes and calculate their properties. By translating geometric problems into algebraic equations, we are able to solve problems more systematically.

The cylinder volume formula \( V = \pi r^2 h \) is a prime example of this intersection:

• The role of algebra here is to provide a means of manipulating this formula to solve for any unknown, such as height \( h \), using algebraic principles.
• Geometric properties, like the dimensions of a cylinder, are expressed in terms of algebraic variables and constants, making it easier to plug in different values and model real-world scenarios.

Understanding this fusion of algebra and geometry empowers students to solve a wider array of problems, enhancing both comprehension and practical application skills.