Determine the Original Radius of a Circle Given an Area Increase

In mathematical problem solving, understanding the relationship between the radius and the area of a circle is fundamental. This article will explore a common problem where an increase in the radius results in a specific increase in the area, and guide you through the process of solving for the original radius step-by-step.

Understanding the Problem

The area of a circle increases by 100 cm2 when its radius is increased by 10 cm. We need to determine the original radius of the circle before the increase. We will present multiple methods to arrive at the solution, ensuring a comprehensive understanding of the problem-solving process.

Solution Method 1

Let the original radius of the circle be (r) cm.

1. Calculate the original area: [ A pi r^2 ]

2. When the radius is increased by 10 cm, the new radius is (r 10) cm. The new area is: [ A' pi (r 10)^2 ]

3. According to the problem, the increase in area is 100 cm2: [ pi (r 10)^2 - pi r^2 100 ]

4. Simplify the equation: [ pi (r^2 20r 100) - pi r^2 100 ]

5. Cancel out the common term (pi r^2): [ pi (20r 100) 100 ]

6. Simplify further: [ 20r 100 frac{100}{pi} ]

7. Solve for (r): [ 20r frac{100}{pi} - 100 ] [ r frac{frac{100}{pi} - 100}{20} ]

8. Using the approximation (pi approx 3.14), calculate the value of (r): [ r approx frac{frac{100}{3.14} - 100}{20} approx frac{31.83 - 100}{20} approx frac{-68.17}{20} approx -3.4 ]

Since a negative radius does not make sense, this solution method indicates an error in the problem setup.

Solution Method 2

Alternatively, use the given information directly and apply algebraic manipulation:

1. Let the original area be (A): [ A pi r^2 ]

2. The new area after the radius is increased by 10 cm: [ A' pi (r 10)^2 ]

3. The increase in area, 100 cm2, can be written as: [ pi (r 10)^2 - pi r^2 100 ]

4. Expand and simplify the equation: [ pi (r^2 20r 100) - pi r^2 100 ] [ pi (20r 100) 100 ]

5. Solve for (r): [ 20r 100 frac{100}{pi} ] [ 20r frac{100}{pi} - 100 ] [ r frac{frac{100}{pi} - 100}{20} ]

6. Using (pi approx 3.14), calculate the value of (r): [ r approx frac{frac{100}{3.14} - 100}{20} approx frac{31.83 - 100}{20} approx frac{-68.17}{20} approx -3.4 ]

Again, this equation suggests an error since a negative value for the radius is not valid.

Solution Method 3

Another approach involves recognizing the quadratic relationship between the radius and the area:

1. The increase in area can be expressed as: [ pi (r 10)^2 - pi r^2 22 ] [ pi (r^2 20r 100) - pi r^2 22 ]

2. Simplify the equation: [ pi (20r 100) 22 ]

3. Solve for (r): [ 20r 100 frac{22}{pi} ] [ 20r frac{22}{pi} - 100 ] [ r frac{frac{22}{pi} - 100}{20} ]

4. Using (pi approx 3.14), calculate the value of (r): [ r approx frac{frac{22}{3.14} - 100}{20} approx frac{7 - 100}{20} approx frac{-93}{20} approx -4.65 ]

Again, a negative radius is not feasible, indicating that there may be an issue with the problem statement.

Conclusion

Each method reveals the consistent negative value for the radius, which doesn’t make sense in the context of the problem. If the problem statement is correct, it suggests a conceptual or arithmetic error in the setup. To obtain real and meaningful results, further verification or corrections to the problem statement would be necessary.