A circular mirror is surrounded by a square metal frame. The radius of the mirror is 2x. The side length of the metal frame is 8x. What is the area of the metal frame? Write your answer in factored form.
(8x)^2 - [π * (2x)^2]
Why did the mirror hire a personal trainer?
Because it wanted to get fit and reflect back its best self!
Now, let's work on the math problem.
The area of the metal frame can be calculated by subtracting the area of the circular mirror from the area of the square frame.
The area of a circle is given by the formula: A = πr^2, where r is the radius. In this case, the radius is 2x, so the area of the circle is A_mirror = π(2x)^2 = 4πx^2.
The area of a square is given by the formula: A = s^2, where s is the side length. In this case, the side length is 8x, so the area of the square is A_frame = (8x)^2 = 64x^2.
Therefore, the area of the metal frame is A_frame - A_mirror = 64x^2 - 4πx^2.
In factored form, the area of the metal frame can be written as (64 - 4π)x^2.
To find the area of the metal frame, we need to subtract the area of the circular mirror from the area of the square frame.
The area of a square is given by A = s^2, where s is the side length.
So, the area of the square metal frame is (8x)^2 = 64x^2.
The area of a circle is given by A = πr^2, where r is the radius.
So, the area of the circular mirror is π(2x)^2 = 4πx^2.
To find the area of the metal frame, we subtract the area of the circular mirror from the area of the square frame:
Area of metal frame = 64x^2 - 4πx^2.
The common factor here is x^2. Factoring it out, we get:
Area of metal frame = x^2(64 - 4π).
Thus, the area of the metal frame, written in factored form, is x^2(64 - 4π).
To find the area of the metal frame, we need to subtract the area of the circular mirror from the area of the square metal frame.
The area of the circular mirror can be found using the formula for the area of a circle, which is given by A = πr², where A is the area and r is the radius. In this case, the radius is 2x, so the area of the circular mirror is A_mirror = π(2x)² = 4πx².
The area of the square metal frame can be found by subtracting the area of the circular mirror from the area of the square frame. The side length of the square frame is 8x, so the area of the square frame is A_frame = (8x)² = 64x².
Therefore, the area of the metal frame is A_frame - A_mirror = 64x² - 4πx².
In factored form, we can factor out the common factor of x² to get (64 - 4π)x².