Problem 90 - Elementary Algebra

Question

A circle has an area of \(25 \pi\) square units. By what amount will the radius have to be increased to create a circle with double the given area?

Step-by-Step Solution

Answer

The radius needs to be increased by approximately 2.07 units.

Step 1: Calculate the Original Radius

The area of a circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius. We are given that the area is \( 25\pi \) square units. Setting up the equation: \( 25\pi = \pi r^2 \). Divide both sides by \( \pi \) to solve for \( r^2 \): \( 25 = r^2 \). Taking the square root of both sides gives \( r = 5 \). So, the original radius is 5 units.

Step 2: Calculate the New Area

We want the new circle to have an area that is double the original area. So, the new area \( A_{new} \) is \( 2 \times 25\pi = 50\pi \) square units.

Step 3: Calculate the New Radius

Using the area formula again, \( A_{new} = \pi R^2 \), where \( R \) is the new radius. Plug in the new area: \( 50\pi = \pi R^2 \). Divide both sides by \( \pi \) to get \( 50 = R^2 \). Taking the square root of both sides gives \( R = \sqrt{50} = 5\sqrt{2} \approx 7.07 \).

Step 4: Find the Increase in Radius

Subtract the original radius from the new radius to find the required increase: \( 5\sqrt{2} - 5 = 5(\sqrt{2} - 1) \). Approximate this result: \( 5(1.414 - 1) \approx 5 \times 0.414 = 2.07 \). Thus, the radius needs to be increased by approximately 2.07 units.

Key Concepts

Circle Area

Understanding the area of a circle is essential in algebra and geometry. A circle's area is the total space enclosed within its boundary. To calculate this, we use the formula:

• The area (\( A \)) is given by the formula \( A = \pi r^2 \), where \( r \) is the circle's radius.

This formula shows how the size of a circle changes with the square of the radius. Notice that \( \pi \) is a constant approximately equal to 3.14159. When you know the area, you can work backwards to find out more about the circle, such as the radius. In the given exercise, with an area of \( 25\pi \) square units, we can determine the radius by rearranging the formula to: \( r^2 = 25 \). Use a calculator or understand that the square root of 25 is 5 to find that the radius is 5 units.

Radius Calculation

Calculating the radius of a circle given its area is critical when solving many geometry problems. The process involves working with the formula \( A = \pi r^2 \) to isolate \( r \).

• First, divide the given area by \( \pi \) to remove the constant from the equation. This gives us \( r^2 = \frac{A}{\pi} \).
• Then, take the square root of both sides to solve for the radius: \( r = \sqrt{\frac{A}{\pi}} \).

In the specific exercise, once you double the area from \( 25\pi \) to \( 50\pi \), the same steps will give you the new radius using \( R^2 = 50 \). Solving this tells us that the new radius is \( 5\sqrt{2} \) or approximately 7.07 units. Finally, subtract the original radius from this new radius to find how much the radius needs to increase.

Geometry

Geometry deals with questions of shape, size, and the properties of space. The circle, a two-dimensional shape where all points are equidistant from a single point (the center), is fundamental to many geometric problems.

• Key properties include the radius, diameter (twice the radius), and area.
• Understanding circles also involves recognizing constant relationships, such as how the area scales with the square of the radius.

The exercise also highlights how changes in one property, like area, directly affect others, like the radius. To enlarge a circle to double its area, you don't just double the radius. Instead, you increase it according to the square root of the scaling factor of the area. Therefore, refining your understanding of relationships between these properties provides a strong foundation in geometry and helps solve complex problems efficiently.