Problem 60 - College Algebra

Question

The circle formed by the middle lane of a circular running track can be described algebraically by \(x^{2}+y^{2}=4\) where all measurements are in miles. If you run around the track's middle lane twice, approximately how many miles have you covered?

Step-by-Step Solution

Answer

If you run around the track's middle lane twice, you will have covered approximately \(8\pi\) or about 25.13 miles.

Step 1: Identify the radius of the circle

The equation of the circle is \(x^{2}+y^{2}=4\). From this equation, we see that the circle's radius \(r^{2}\) equals 4. Therefore, \(r = \sqrt{4} = 2\) miles.

Step 2: Calculate the circumference of the circle.

Use the formula for a circle's circumference, which is \(2\pi r\). For this circle, it would be \(2\pi \times 2 = 4\pi\) miles.

Step 3: Calculate the total distance covered.

Since the question says that you run around twice, you will double the calculated distance. So the total distance you run is \(2 \times 4\pi = 8\pi\) miles.

Key Concepts

Circle Equations

Understanding circle equations is crucial in geometry. A common way to represent a circle on the Cartesian plane is using the equation

• \(x^{2} + y^{2} = r^{2}\), which describes a circle with its center at the origin \((0, 0)\).
• Here, \(x\) and \(y\) are variables representing points on the circle circumference.
• The term \(r^{2}\) is the square of the radius.

In the context of our original exercise, we have

• the equation \(x^{2} + y^{2} = 4\).

This tells us two things:

• First, the circle's center is at the origin.
• Second, the radius squared is 4, which means the radius \(r\) is 2 miles.

These insights are fundamental to solving any geometric problems involving circles.

Radius Calculation

The radius of a circle is key to many geometric calculations. It tells us how far any point on the circumference is from the circle's center. For the circle described by the equation \(x^{2} + y^{2} = r^{2}\), the calculation for finding the radius \(r\) involves extracting the square root of the constant on the right side of the equation.

• In our case, \(x^{2} + y^{2} = 4\) implies \(r^{2} = 4\).
• Thus, \(r = \sqrt{4} = 2\).

The radius is integral as it is used to calculate other vital attributes like the circumference. The value \(r = 2\) miles not only helps identify the circle's dimensions but also allows us to compute the distance run by performing simple multiplication.

Distance Calculation

Distance calculation on a circular track is often related to finding the circle's circumference. The circumference \(C\) of the circle is calculated by the formula

• \(C = 2\pi r\).

By substituting \(r = 2\) miles (as found in the radius calculation), we get

• \(C = 2\pi \times 2 = 4\pi\) miles.

This is the distance covered when running around the circle once. However, if you remember the problem, it stated that you run around the track twice. Therefore, you double the circumference:

• Total distance = \(8\pi\) miles.

This calculation highlights not only the importance of understanding circle equations but also emphasizes practical applications, such as determining the exact track distance for training and athletic purposes.