An artist framed a rectangular picture. The dimensions of the frame are 4x+3 and 3x+5. He is going to place the picture with the border of 2 inches. Then he wants to paint the border. What is the area he has to paint?
area of picture = (4x+3)(3x+5)
= 12x^2 + 29x + 15
area of whole thing = (4x+3 +4)(3x+5 + 4)
= (4x+7)(3x+9)
= 12x^2 + 57x+63
area of border
= 12x^2 + 57x+63 - (12x^2 + 29x + 15)
= 28x + 48
To find the area of the frame that needs to be painted, we first need to find the dimensions of the picture itself.
The dimensions of the frame are given as 4x+3 and 3x+5. Since the border is 2 inches on all sides, we need to subtract twice the border width from each dimension to get the dimensions of the picture.
So, the dimensions of the picture will be (4x+3) - 2(2) and (3x+5) - 2(2), which simplifies to 4x - 1 and 3x + 1.
To calculate the area of the frame, we need to find the area of the outer rectangle and subtract the area of the inner rectangle (the picture). The formula for the area of a rectangle is length x width.
The area of the outer rectangle, which is the frame, is (4x+3) x (3x+5).
The area of the inner rectangle, which is the picture, is (4x - 1) x (3x + 1).
Now, to find the area that needs to be painted, we subtract the area of the picture from the area of the frame:
(4x+3)(3x+5) - (4x - 1)(3x + 1)
You can simplify this expression by multiplying out the terms and then combining like terms.
Once simplified, the resulting expression will give you the area of the frame that needs to be painted.