Problem 58 - Algebra and Trigonometry

Question

Solve the equation for the indicated variable. \(A=2 \pi r^{2}+2 \pi r h ; \quad\) for \(r\)

Step-by-Step Solution

Answer

The equation for \( r \) is solved using the quadratic formula.

Step 1: Identify the equation and the variable

The given equation is \( A = 2\pi r^2 + 2\pi rh \). We are required to solve for \( r \), which means we must isolate \( r \) on one side of the equation.

Step 2: Factor out common terms

Notice that both terms on the right-hand side have a common factor of \( 2\pi r \). Factor \( 2\pi r \) from the expression to simplify: \[ A = 2\pi r (r + h) \]

Step 3: Solve for r

To solve for \( r \), divide both sides of the equation by \( 2\pi (r + h) \): \[ r = \frac{A}{2\pi (r + h)} \] However, we encounter \( r \) on both sides, so the equation cannot be solved simply by isolation. Rearranging might not yield an explicit solution for \( r \). Typically, numerical methods might be the next approach as this is a quadratic form in \( r \).

Step 4: Express as quadratic equation

Reorganize the equation: \[ A = 2\pi r^2 + 2\pi rh \] This translates into a quadratic expression: \[ 2\pi r^2 + 2\pi rh - A = 0 \] Treat this as a quadratic equation in terms of \( r \): \[ ar^2 + br + c = 0 \] where \( a = 2\pi \), \( b = 2\pi h \), and \( c = -A \).

Step 5: Apply the quadratic formula

The quadratic formula is given by: \[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute for \( a = 2\pi \), \( b = 2\pi h \), and \( c = -A \) in the formula, leading to: \[ r = \frac{-2\pi h \pm \sqrt{(2\pi h)^2 - 4 \cdot 2\pi \cdot (-A)}}{2 \cdot 2\pi} \] Simplify the radical and the entire expression to solve for \( r \).

Step 6: Simplify the expression

Calculate within the square root and simplify: \[ r = \frac{-2\pi h \pm \sqrt{4\pi^2 h^2 + 8\pi A}}{4\pi} \] Finally, this gives you potential solutions for \( r \).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are mathematical expressions that include a variable raised to the power of two. They appear in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable we are solving for. In everyday terms, a quadratic equation looks like a U-shaped curve on a graph, a parabola, which can open upwards or downwards. These equations are important because they appear in various practical scenarios, such as physics for calculating projectile motion or economics to find maximum profit. Understanding and solving these equations is a cornerstone of algebra.

The Art of Factoring

Factoring is a technique used to simplify equations, making them easier to solve. It involves breaking down a composite number or expression into a product of smaller or simpler terms, known as factors. For instance, with the equation \( A = 2\pi r (r + h) \), you notice both terms on the right include \( 2\pi r \), allowing you to factor it out. By rewriting equations in their factorized form, you often discover common roots or can simplify complex equations. Factoring is essential in solving polynomial equations, particularly because it can help make complicated calculations much more manageable through simplification.

The Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). It provides a straightforward plug-and-go method to find the values of \( x \). The formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Each part of this formula is crucial.

• \( b^2 - 4ac \) is called the discriminant and determines the nature of the roots.
• A positive discriminant means two real solutions, while a zero discriminant indicates one real solution.
• If negative, the equation has no real solutions.

The quadratic formula works universally on all quadratic equations, making it particularly useful when factoring is not easily applicable or practical.

Simplification of Mathematical Expressions

Mathematical expression simplification is the process of reducing an expression to its simplest form, without changing its value. This involves combining like terms, canceling terms, and applying mathematical operations to present the expression as clearly and compactly as possible. Take for example the expression: \[ r = \frac{-2\pi h \pm \sqrt{4\pi^2 h^2 + 8\pi A}}{4\pi} \]Here, simplification would mean performing operations within the square root, removing common factors, and rewriting where necessary to make it more digestible and ready for further calculations. Simplification can significantly reduce the complexity of solving equations, making it a key step in mathematical problem-solving.