Question

b. noah drew a circle with an area of 500 in². han drew a circle with a diameter that is 3 times the diameter of noahs circle. estimate the area of hans circle. explain your reasoning.

Explanation:

Step1: Recall the area formula for a circle

The area formula for a circle is \(A = \pi r^{2}\), where \(r\) is the radius. If the diameter \(d = 2r\), and if the diameter of one circle is \(k\) times the diameter of another circle, the radius of the first circle is also \(k\) times the radius of the second circle. Let the radius of Noah's circle be \(r_1\) and its area \(A_1=\pi r_1^{2}=500\) in². The diameter of Han's circle \(d_2 = 3d_1\), so the radius of Han's circle \(r_2=3r_1\).

Step2: Find the area of Han's circle

The area of Han's circle \(A_2=\pi r_2^{2}\). Substitute \(r_2 = 3r_1\) into the formula: \(A_2=\pi(3r_1)^{2}\). Using the exponent rule \((ab)^{n}=a^{n}b^{n}\), we get \(A_2=\pi\times9r_1^{2}\). Since \(A_1 = \pi r_1^{2}=500\) in², then \(A_2 = 9\times A_1\).

Step3: Calculate the area

Substitute \(A_1 = 500\) in² into the equation \(A_2=9\times A_1\). So \(A_2=9\times500=4500\) in².

Answer:

The area of Han's circle is \(4500\) in².