Problem 53 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 51-54: Solve for the specified variable. $$ A=2 \pi r(r+h) \text { for } r $$ (surface area of a closed cylinder)

Step-by-Step Solution

Answer

\( r = \frac{-h + \sqrt{h^2 + \frac{2A}{\pi}}}{2} \).

Step 1: Understanding the Problem

We are given the equation for the surface area of a closed cylinder, which is \( A = 2 \pi r (r+h) \). The task is to solve this equation for the variable \( r \). This means we need to express \( r \) in terms of \( A \) and \( h \).

Step 1: Emphasizing the Equation

Start by writing down the equation: \[ A = 2 \pi r (r + h) \].Our goal is to isolate \( r \) on one side of the equation.

Step 2: Divide by the Common Factor

To simplify the equation, divide both sides by \( 2\pi \):\[ \frac{A}{2\pi} = r (r + h) \].This expression makes it easier to deal with the terms involving \( r \).

Step 3: Quadratic Equation Formulation

We see that \( r (r + h) \) can be expanded as a quadratic equation: \( r^2 + hr \).Rewrite the equation as:\[ r^2 + hr - \frac{A}{2\pi} = 0 \].This is a standard quadratic equation in the form \( ar^2 + br + c = 0 \), where \( a = 1 \), \( b = h \), and \( c = -\frac{A}{2\pi} \).

Step 4: Applying the Quadratic Formula

To solve the quadratic equation, use the quadratic formula:\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].Substitute \( b = h \), \( a = 1 \), and \( c = -\frac{A}{2\pi} \) into the formula:\[ r = \frac{-h \pm \sqrt{h^2 + 4\cdot\frac{A}{2\pi}}}{2}\].

Step 5: Simplifying the Expression

Simplify the discriminant part:\[ r = \frac{-h \pm \sqrt{h^2 + \frac{2A}{\pi}}}{2}\].This expression gives the possible values for \( r \). Since \( r \) represents a radius, it must be non-negative, so choose the positive root.

Key Concepts

Surface Area

When discussing the surface area of a closed cylinder, it's important to understand what this term means. The surface area is the total area that the surface of an object occupies. In the case of a closed cylinder, this includes the area of both circular ends as well as the rectangular area wrapped around the sides.

The formula for the surface area of a closed cylinder is given by \(A = 2\pi r(r + h)\) where:

• \(A\) is the total surface area of the cylinder.
• \(r\) is the radius of the cylinder's base.
• \(h\) is the height of the cylinder.

This formula arises from the sum of the area of two circles \(2\pi r^2\) (one for each end) plus the area of the rectangle that wraps around the cylinder \(2\pi rh\). Understanding this composition is crucial when manipulating the equation to solve for specific variables, like the radius \(r\).

Closed Cylinder

A closed cylinder is a three-dimensional shape with two parallel bases that are congruent circles and a curved surface that connects these bases. Cylinders are common in daily life and engineering, appearing in objects like cans, pipes, and barrels.

When we say a cylinder is 'closed,' it means that there are no openings; both bases are covered. This makes a difference in calculations because the surface area now includes the top and bottom circles in addition to the side. This is different from an 'open' cylinder, which would lack one or both ends.

In terms of geometrical properties, consider these key points:

• The distance between the two bases is the height \(h\) of the cylinder.
• The distance from the center to any point on the base edge is the radius \(r\).

Remembering these structural details helps clarify why the surface area formula was derived as it is, ensuring accuracy in applications involving cylinders.

Quadratic Formula

The quadratic formula is a powerful tool for solving equations in the form \(ax^2 + bx + c = 0\). It's specifically used for quadratic equations, which are polynomials of degree 2.

The general formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where:

• \(a\), \(b\), and \(c\) are coefficients from the equation \(ax^2 + bx + c = 0\).
• \(\pm\) indicates that there are generally two solutions for \(x\).
• \(\sqrt{b^2 - 4ac}\) is called the discriminant.

The discriminant determines the nature of the roots: if it's positive, there are two distinct real roots; if zero, one real root; and if negative, two complex roots. When solving quadratic equations like the one for the cylinder's radius, identifying the correct roots based on context allows us to understand which solutions make sense physically, like ensuring a non-negative radius.

Problem Solving Steps

Approaching a mathematical problem systematically can simplify even complex tasks. Here’s how to best tackle such problems, particularly those involving equations like the surface area of a closed cylinder:

Begin with Understanding:

• Identify the given information and the unknown variable. Translate the word problem into mathematical language if needed.
• Make sure the equation is correctly noted from the problem statement.

Implement Problem-Specific Strategies:

• Look for techniques to simplify like dividing both sides by constants or refactoring terms to reveal hidden patterns.
• For quadratic expressions, formulating them properly is crucial. Recognize what a quadratic expression looks like to use the quadratic formula efficiently.

Execute and Verify:

• After applying the correct formulas and techniques, ensure your solution is logical for the problem context (e.g., a radius needs to be non-negative).
• Verify by plugging the solution back into the original equation to check for accuracy, ensuring no steps have been overlooked.

Always keep a checklist of these steps when faced with challenges, and your problem-solving will improve over time.