Please explain to me how to solve:
A circular mirror is surrounded by a square metal frame. The radius of the mirror is 5x. The side length of the metal frame is 15x. What is the area of the metal frame? Write your answer in factored form.
Formula for area of frame minus formula for area of mirror is the REMAINING area of frame.
Area of frame = (15x)^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.
1. Start by finding the area of the circular mirror:
The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.
Given that the radius of the mirror is 5x, we can substitute it into the formula:
A_mirror = π * (5x)^2 = 25πx^2.
2. Next, find the area of the square frame:
The formula for the area of a square is A = s^2, where A is the area and s is the side length.
Given that the side length of the metal frame is 15x, we can substitute it into the formula:
A_frame = (15x)^2 = 225x^2.
3. Subtract the area of the circular mirror from the area of the square frame:
Area_of_frame = A_frame - A_mirror
= 225x^2 - 25πx^2.
4. Factoring out the common factor of x^2, we get:
Area_of_frame = x^2 (225 - 25π).
Therefore, the area of the metal frame is x^2 (225 - 25π).