Problem 89 - Algebra for College Students

Question

Set up an equation and solve each of the following problems. The area of a circular region is numerically equal to three times the circumference of the circle. Find the length of a radius of the circle.

Step-by-Step Solution

Answer

The radius of the circle is 6 units.

Step 1: Understanding the Problem

We need to find the radius of a circle given that its area is equal to three times its circumference.

Step 2: Define the Formulas

The formula for the area of a circle is \( A = \pi r^2 \). The formula for the circumference of a circle is \( C = 2\pi r \).

Step 3: Set Up the Equation

We know from the problem statement that the area is equal to three times the circumference: \( \pi r^2 = 3(2\pi r) \).

Step 4: Simplify the Equation

Simplifying the equation \( \pi r^2 = 6\pi r \) by dividing both sides by \( \pi \) (assuming \( r eq 0 \)) gives \( r^2 = 6r \).

Step 5: Solve the Quadratic Equation

Subtract \( 6r \) from both sides to get \( r^2 - 6r = 0 \). Factor the equation: \( r(r - 6) = 0 \). The solutions are \( r = 0 \) or \( r = 6 \).

Step 6: Consider the Practical Solution

Since a radius of 0 doesn't make sense for a circle, the radius must be \( r = 6 \).

Key Concepts

Circle Area

The area of a circle is a measure of how much space is enclosed within the circle. It can be thought of as the total number of square units that fit inside. To find the area, we use the formula:- The formula: \( A = \pi r^2 \)- \( \pi \) is a special constant that is approximately equal to 3.14159, representing the ratio of a circle's circumference to its diameter.- \( r \) is the radius, the distance from the center of the circle to any point on its edge. For example, if the radius of a circle is 3 units, the area is calculated as \( A = \pi (3)^2 = 9\pi \). If you fill in \( \pi \), the area is approximately 28.27 square units.Understanding the area formula helps you solve problems by plugging in different values for the radius, allowing you to easily compute areas for circles with different sizes.

Circumference of a Circle

The circumference of a circle is the distance around the circle, much like the perimeter of a polygon. You can use the formula:- The formula: \( C = 2\pi r \)- \( \pi \) is the same constant that we use for circle-related formulas, about 3.14159.- \( r \) is the radius, which is half the diameter. If you have a circle with a radius of 4 units, the circumference is \( C = 2\pi(4) = 8\pi \). This translates to an approximate length of 25.13 units.Knowing how to calculate the circumference is essential in problems where you need to understand the size of the boundary or outline of a circle compared to its enclosed area or when these quantities relate to each other, like in the given problem.

Quadratic Equations

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \), where "a," "b," and "c" are constants. These equations often have two solutions. Let's break down the steps:- When the equation is simplified into \( r^2 = 6r \), it becomes quadratic by transforming it to \( r^2 - 6r = 0 \).- The quadratic formula or factoring can solve it. Here, factoring is used. Factoring involves looking for values that multiply to form "a*c" and add to give "b". The given problem factors cleanly as \( r(r - 6) = 0 \), so:- Solutions are found by setting each factor equal to zero: \( r = 0 \) or \( r = 6 \). The practical solution is \( r = 6 \), since a zero radius isn't realistic for a physical circle. Understanding quadratic equations allows solving a broad range of problems like optimizing objects, analyzing trajectories, or understanding geometrical properties.