Problem 68 - Algebra and Trigonometry with Analytic Geometry

Question

The boundary of a city is a circle of diameter 10 miles. Within the last decade, the city has grown in area by approximately \(16 \pi \mathrm{mi}^{2}\) (about \(50 \mathrm{mi}^{2}\) ). Assuming the city was always circular in shape, find the corresponding change in distance from the center of the city to the boundary.

Step-by-Step Solution

Answer

The change in radius is approximately 1.4 miles.

Step 1: Understand Initial Conditions

The problem states that the city is initially circular with a diameter of 10 miles. This means the initial radius is half of the diameter, i.e., 5 miles. The area of the initial circle is calculated using the formula for the area of a circle, \(A = \pi r^2\).

Step 2: Calculate Initial City Area

Using the initial radius, \(r = 5\), the initial area of the city can be calculated as follows: \(A_1 = \pi (5)^2 = 25\pi\ \text{mi}^2\).

Step 3: Calculate Final City Area

The new area of the city is given as having increased by \(16\pi\ \text{mi}^2\). Therefore, the final area is \(A_2 = 25\pi + 16\pi = 41\pi\ \text{mi}^2\).

Step 4: Determine Final Radius

Using the formula for the area of a circle, \(A = \pi r^2\), and the final area, solve for the final radius. Set \(41\pi = \pi r^2\), which simplifies to \(r^2 = 41\). The final radius is \(r = \sqrt{41}\).

Step 5: Calculate Change in Radius

The change in the distance from the center of the city to its boundary is the difference between the final radius and the initial radius. \(\Delta r = \sqrt{41} - 5\).

Step 6: Simplify Change in Radius

Compute the change in radius: \(\sqrt{41} \approx 6.4\), hence, \(\Delta r = 6.4 - 5 = 1.4\).

Step 7: Conclusion

The city's boundary expanded radially outward by approximately 1.4 miles.

Key Concepts

Radius

The radius is a key concept when dealing with circles of any kind, including geometrical shapes like the city discussed in this exercise. The radius of a circle is defined as the distance from the center of the circle to any point on its boundary. It is denoted by the letter \(r\). This value is crucial because it helps us calculate other important characteristics of the circle, such as its area and circumference.

Calculating the Radius

• Given the diameter of a circle (which is the longest distance from one side of the circle to the other through the center), the radius is simply half of the diameter.
• In this exercise, the city's initial diameter is 10 miles. Therefore, the initial radius is \(5\) miles, since \( \text{radius} = \frac{\text{diameter}}{2} \).

Area of a Circle

The area of a circle is an important concept in geometry. It helps in understanding the space enclosed within the circle's boundary. The formula to calculate the area of a circle is given by \(A = \pi r^2\), where \(A\) represents the area and \(r\) the radius of the circle.

Understanding Circle Area

• The area can be thought of as the number of square units that fit inside the circle.
• Initially, the city had an area of \(25\pi\) square miles, since the initial radius was \(5\) miles.
• When the area grows, as in our exercise, you can use the change in area to find the new radius of the circle.

The formula is useful not only in finding out how large a circle is but also in solving various real-life problems like city planning and land utilization.

Distance from Center to Boundary

This is essentially the radius of the circle and represents how far you would travel from the center of a circle to any point on its perimeter. This distance changes when the circle grows or shrinks.

Calculating Distance Change

• In our exercise, the city's boundary expanded, meaning its radius changed. Initially, the radius was 5 miles.
• After the city grew larger, the final radius became \(\sqrt{41}\), which is approximately \(6.4\) miles.
• The change in distance from the center to the boundary is the difference between the final and initial radius, calculated to be approximately 1.4 miles.

Understanding this concept is key in problems involving growth or reduction of circular areas, providing insightful information about how and by how much such changes occur.