Problem 62 - Algebra for College Students

Question

Set up an equation and solve each problem. The sum of the areas of two circles is \(65 \pi\) square feet. The length of a radius of the larger circle is 1 foot less than twice the length of a radius of the smaller circle. Find the length of a radius of each circle.

Step-by-Step Solution

Answer

The radii are 4 feet (smaller circle) and 7 feet (larger circle).

Step 1: Identify Variables

Let the radius of the smaller circle be \( r \). Then, the radius of the larger circle, according to the problem statement, is \( 2r - 1 \).

Step 2: Write Area Formulas

The area of the smaller circle is \( \pi r^2 \). The area of the larger circle is \( \pi (2r - 1)^2 \).

Step 3: Set Up the Equation

According to the problem, the sum of the two areas is \( 65\pi \). Hence, we have: \[ \pi r^2 + \pi (2r - 1)^2 = 65\pi \] Simplify to get: \[ r^2 + (2r - 1)^2 = 65 \]

Step 4: Expand and Combine Terms

Expand the quadratic expression: \[ (2r - 1)^2 = 4r^2 - 4r + 1 \] So the equation becomes: \[ r^2 + 4r^2 - 4r + 1 = 65 \] Simplify to: \[ 5r^2 - 4r + 1 = 65 \]

Step 5: Formulate the Quadratic Equation

Move all terms to one side to get:\[ 5r^2 - 4r + 1 - 65 = 0 \] Simplify to:\[ 5r^2 - 4r - 64 = 0 \]

Step 6: Solve the Quadratic Equation

Use the quadratic formula where \( a = 5 \), \( b = -4 \), and \( c = -64 \). The quadratic formula is: \[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute values: \[ r = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(5)(-64)}}{2(5)} \] Simplify inside the square root: \[ r = \frac{4 \pm \sqrt{16 + 1280}}{10} \] \[ r = \frac{4 \pm \sqrt{1296}}{10} \] Since \( \sqrt{1296} = 36 \), plug it back in: \[ r = \frac{4 \pm 36}{10} \] So, \( r = 4 \) (choosing the positive solution).

Step 7: Find Radii

Since the radius of the smaller circle is \( r = 4 \), for the larger circle: \[ 2r - 1 = 2(4) - 1 = 7 \] Hence, the radii are 4 feet and 7 feet for the smaller and larger circles, respectively.

Key Concepts

Quadratic Equations

Quadratic equations often appear in algebra word problems, especially when dealing with areas and lengths. These are equations of the form \( ax^2 + bx + c = 0 \). In these cases, identifying the variables correctly is crucial. Solving such equations typically involves a few steps:

• First, set up the equation accurately from the problem statement.
• Then, simplify by expanding any expressions.
• Move all terms to one side to obtain a standard form.
• Finally, use methods like the quadratic formula to find solutions.

In the given problem, the equation formed was \( 5r^2 - 4r - 64 = 0 \). To solve, we used the quadratic formula: \[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]By substituting the appropriate coefficients \((a = 5, b = -4, c = -64)\), we were able to find the possible values for \(r\). Because problem contexts often imply that only positive solutions are viable, we chose \(r = 4\) as the valid answer.

Circle Geometry

Circle geometry is essential for understanding problems involving circles, such as calculating areas. Recall that the formula for the area of a circle is: \[ \text{Area} = \pi r^2\]where \(r\) is the radius of the circle. Understanding this concept helps when you need to express areas in terms of algebraic equations.In this particular problem, you needed to express the areas of two circles. The area of the smaller circle was \( \pi r^2 \), and for the larger circle, it was \( \pi (2r - 1)^2 \). These expressions were summed and equated to \(65\pi\) since the total given was \(65\pi\). Identifying these area expressions correctly is key when breaking down the problem.

Problem Solving Steps

Effectively solving algebra word problems requires a systematic approach:

• Identify Variables: Begin by defining the variables, using the problem's conditions.
• Translate Words to Equations: Carefully convert the situation into a mathematical equation.
• Simplify and Reduce: Use algebraic techniques to simplify the equation for easier manipulation.
• Solve and Interpret: Once solved, interpret the solutions in the context of the problem.
• Check Your Answer: Substitute solutions back into the original context to verify accuracy.

In our example, the problem was solved by following these steps. The radii were identified and related through a logical expression \((2r - 1)\). The equation was set up, simplified, solved using the quadratic formula, and the results were verified to ensure they fitted the problem conditions: radii of \(4\) feet and \(7\) feet.