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.
25x^2(9-pi)
To solve this problem, we need to find the area of the metal frame surrounding the circular mirror.
First, let's find the area of the circular mirror. The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius. In this case, the radius of the mirror is 5x. So, the area of the mirror is:
A_mirror = π(5x)^2
Simplifying this further, we get:
A_mirror = π(25x^2) = 25πx^2
Now, let's find the area of the square metal frame. The area of a square is given by the formula A = s^2, where A is the area and s is the side length. In this case, the side length of the metal frame is 15x. So, the area of the frame is:
A_frame = (15x)^2
Simplifying this further, we get:
A_frame = 225x^2
Finally, to find the area of the metal frame, we need to subtract the area of the mirror from the area of the frame:
A_metal_frame = A_frame - A_mirror
Substituting the values we found earlier, we have:
A_metal_frame = 225x^2 - 25πx^2
So, the area of the metal frame is 225x^2 - 25πx^2 in factored form.
frame= 15x * 15x = 225x^2
mirror= pi(r)^2 = pi(25x)^2
area of metal(answer in factored form)=225x^2-(25)pi x2 = 25x^2(25-pi)
area of square = 15x*15x = 225 x^2
area of mirror = pi (25) x^2
area of metal = 225 x^2 - 25 pi x^2
= 25 x^2 (25 -pi)