Problem 23 - Algebra A Combined Function

Question

Solve each formula for the specified variable. \(S=2 \pi r h+2 \pi r^{2}\) for \(h\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Equation

The given formula is related to the surface area of a cylinder:\[ S = 2 \pi r h + 2 \pi r^2 \]Our task is to solve this equation for \( h \).

Step 2: Isolate the Term Involving h

Subtract the term not involving \( h \) from both sides to get the term with \( h \) alone:\[ S - 2 \pi r^2 = 2 \pi r h \]

Step 3: Solve for h

Divide both sides of the equation by \( 2 \pi r \) to solve for \( h \):\[ h = \frac{S - 2 \pi r^2}{2 \pi r} \]

Key Concepts

Formulas in Geometry

Geometry uses specific formulas to calculate measurements related to shapes. These formulas are essential for determining various properties, such as perimeter, area, volume, and surface area. Each geometric shape, like a square or a circle, has its own set of formulas. For example,

• The formula for the area of a rectangle is length times width: \( A = l \times w \).
• The circumference of a circle is given by \( C = 2 \pi r \), where \( r \) is the radius.

Using these formulas helps students solve problems related to shapes precisely. Next time you come across a shape problem, remember: It all starts with the right formula!

Surface Area of a Cylinder

A cylinder's surface area includes the areas of its two circular bases and the curved surface that connects these bases. The formula to calculate it is: \[ S = 2 \pi r h + 2 \pi r^2 \]. Here,

• The term \( 2 \pi r h \) represents the curved surface area.
• The term \( 2 \pi r^2 \) accounts for the two bases (top and bottom).

Understanding this formula is important for practical reasons, like finding out how much material is needed to wrap a cylindrical object. Visualize the cylinder as a can with a wrap-around label (the curved area) and lids (the circular ends), all of which contribute to the total surface area.

Isolating Variables

Isolating variables means getting a specific variable by itself on one side of the equation. This is crucial when solving for unknowns. In our exercise, the goal was to extract \( h \) from the equation:

• First, identify terms not involving \( h \) and move them to the other side using subtraction. This simplifies the equation.
• Next, divide the remaining terms by the coefficient linked to \( h \), which in this case is \( 2 \pi r \).

Once isolated, the equation clearly expresses \( h \): \[ h = \frac{S - 2 \pi r^2}{2 \pi r} \]. Practicing this technique enhances problem-solving skills and understanding of mathematical operations.