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The surface area of a cylinder is a crucial concept in geometry and physics, as it defines the total area that covers the cylindrical surface. A cylinder is composed of two circular bases (top and bottom) and a curved surface that connects them. To calculate its surface area, we use the formula: \[ S = 2\pi rh + 2\pi r^2 \] where:

• \( S \) is the surface area.
• \( r \) represents the radius of the circular base.
• \( h \) stands for the height of the cylinder.
• \( 2\pi rh \) corresponds to the lateral or curved surface area.
• \( 2\pi r^2 \) is the combined area of the two bases.

Understanding these components helps us manipulate the equation when solving for different variables or calculating dimensions based on given measurements.

When working with equations, isolating variables is a key mathematical skill. It involves rearranging the equation to express one specific variable in terms of others. This step is crucial for problem-solving and understanding relationships between variables. For the given equation of a cylinder's surface area:\[ S = 2\pi rh + 2\pi r^2 \]we need to solve for the radius \( r \). The process involves:

• Factoring common terms, such as \( 2\pi r \), to simplify the equation.
• Rearranging terms to isolate the variable \( r \) on one side of the equation.
• Dividing and simplifying expressions to further clarify the isolated variable.

By isolating the variable, we make it easier to solve for \( r \) using known values of \( S \) and \( h \), which aids in practical applications.

The quadratic formula is a tool used to find the solutions of quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). It is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula is essential for problems where the equation cannot be factored easily. In the context of our cylinder's surface area equation, when isolated for the radius, it transforms into a quadratic equation: \[ r^2 + hr - \frac{S}{2\pi} = 0 \] Here, \( a = 1 \), \( b = h \), and \( c = -\frac{S}{2\pi} \). Using the quadratic formula allows us to find the value of \( r \), which would satisfy the equation given known values of surface area \( S \) and height \( h \). Remember, practically, we are interested in the positive value of \( r \), as it represents a physical dimension in this context.

Mathematical problem solving is a process where we use logic and reasoning to navigate through a problem. It involves understanding the problem, developing a strategy, carrying out the plan, and then evaluating the solution. In solving the problem of a cylinder's surface area for the radius \( r \), we:

• Identified the initial problem and variable to solve for.
• Utilized strategies such as factoring and rearranging formulas to simplify our approach.
• Applied the quadratic formula to handle the complexity of the quadratic equation.
• Interpreted the solution in the context of a real-world scenario, ensuring the physical meaning makes sense.

Good problem-solving not only solves the immediate challenge but also improves our understanding and ability to handle similar problems in the future.