Problem 46 - Algebra 2

Question

Juwan says that the circle with equation \((x-4)^{2}+y^{2}=36\) has radius 36 units. Lucy says that the radius is 6 units. Who is correct? Explain your reasoning.

Step-by-Step Solution

Answer

Lucy is correct; the radius is 6 units.

Step 1: Understand the Standard Circle Equation

Recall that the standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Identify the Circle Components

Given equation is \((x-4)^2 + y^2 = 36\). Comparing it with the standard form, we identify that \((h, k) = (4, 0)\) and \(r^2 = 36\).

Step 3: Find the Radius from the Equation

From \(r^2 = 36\), we find the radius by taking the square root: \(r = \sqrt{36}\).

Step 4: Calculate the Square Root

Calculate \(\sqrt{36}\), which gives us \(r = 6\).

Step 5: Compare with Given Answers

Compare the calculated radius \(r = 6\) with Juwan's claim that the radius is 36 and Lucy's claim that it is 6. Lucy's claim is correct.

Key Concepts

Standard Form of a Circle

When we talk about the standard form of a circle, we refer to the mathematical way of representing a circle using an equation. The standard form is \[(x-h)^2 + (y-k)^2 = r^2\]where:

• \((h, k)\) is the center of the circle. This tells you the horizontal and vertical positions of the circle on a graph.
• \(r\) is the radius, which is the distance from the center to any point on the circle.

In the given equation, \((x-4)^2 + y^2 = 36\), it's clear to see that the center of the circle is at \((4, 0)\). This equation matches the standard form because it highlights the center and allows us to find the radius by understanding the mathematical components involved.

Radius of a Circle

The radius of a circle is a key concept when dealing with circle equations. In the standard form of a circle, the term on the right side \(r^2\) is the square of the radius. To find the actual radius \(r\), you need to take the square root of that number. Let's look at the given equation \((x-4)^2 + y^2 = 36\). Here, \(36\) is \(r^2\). Taking the square root of \(36\):

• \[ r = \sqrt{36} \]
• This calculates to \(r = 6\)

Thus, the radius of the circle is \(6\) units. Understanding this simple extraction is essential because it helps to interpret circle equations correctly, avoiding the confusion seen with Juwan’s original claim.

Square Root Calculation

The square root calculation is an essential mathematical process when working with circle equations, especially to find the radius. When you're given a radius squared, the next step is to determine what that original radius is by applying the square root function. In our circle equation \[(x-4)^2 + y^2 = 36\]we noted that \(r^2 = 36\). Calculating the square root, we use:

• \(\sqrt{36} = 6\)

This tells us that our radius \(r = 6\). Calculating square roots can involve different methods:

• For perfect squares, like \(36\), the result is a whole number \(6\).
• For non-perfect squares, you may need to use a calculator to find a more precise decimal value.

The square root calculation is crucial in ensuring you determine the circle's dimensions correctly. Understanding how to efficiently compute this prepares you for more complex mathematical challenges.