Problem 68 - A Graphical Approach to College Algebra

Question

Solve each formula for the specified variable.} \(S=2 \pi r h+2 \pi r^{2}\) for \(h \quad\) (Surface area of a cylinder)

Step-by-Step Solution

Answer

\(h = \frac{S - 2 \pi r^2}{2 \pi r}\)

Step 1: Identify the Formula

The given equation represents the surface area of a cylinder, given by the formula: \[ S = 2 \pi r h + 2 \pi r^2 \] We are required to solve this equation for the variable \(h\).

Step 2: Isolate the Term Containing h

Move the term that doesn't involve \(h\) to the other side of the equation by subtracting \(2 \pi r^2\) from both sides. This gives us: \[ S - 2 \pi r^2 = 2 \pi r h \]

Step 3: Solve for h

Divide both sides of the equation by \(2 \pi r\) to isolate \(h\): \[ h = \frac{S - 2 \pi r^2}{2 \pi r} \] This expression gives the value of \(h\) in terms of \(S\) and \(r\).

Key Concepts

Solving Equations

When solving equations, the primary goal is to find the value of the unknown variable. This often involves performing operations that simplify the equation to isolate the variable on one side. The operations we use include addition, subtraction, multiplication, and division.

To visualize this, imagine you have a balanced scale. If you add or remove weight from one side, you must do the same to the other side to keep it balanced. Similarly, in our example, when isolating the term that contains the variable \(h\), we subtracted \(2\pi r^2\) from both sides of our equation. This ensures the equation stays balanced and equivalent to the original.

• Identify what needs to be solved.
• Perform inverse operations to isolate the variable.

Mastering these steps paves the way for solving more complex equations.

Surface Area of a Cylinder

The surface area of a cylinder is an essential concept in geometry, crucial for tasks involving physical measurements. A cylinder has two components that contribute to its surface area: the lateral (side) surface and the two circular bases.

The formula for the surface area requires understanding these elements. The lateral surface area is calculated using the formula \(2\pi rh\), where \(r\) is the radius and \(h\) is the height. The area of the two circular bases is \(2\pi r^2\), combining for a total surface area of \(S = 2\pi rh + 2 \pi r^2\).

• The formula gives us the total covered area.
• It’s useful in real-world problems like paint coverage.

Having this formula handy allows you to plan and evaluate practical scenarios, such as designing containers or measuring materials used in manufacturing.

Formula Manipulation

Formula manipulation is a powerful tool that allows you to rearrange equations to find specific variables, making problem-solving much more flexible. When given a formula, you may need to rearrange it to solve for a particular variable of interest, like the height of a cylinder in our example.

By understanding the structure of the equation, we can strategically move terms. For instance, in the formula \(S = 2\pi rh + 2\pi r^2\), our task was to solve for \(h\). This involved isolating \(h\) by first removing the \(2\pi r^2\) term on the right, followed by dividing each side by \(2\pi r\).

• This process is similar to solving for \(x\) in a basic algebra equation.
• It's about rearranging components logically to highlight the needed variable.

With these skills, you can confidently tackle a wide range of mathematical problems, ensuring clarity and understanding in varied contexts.