a square painting is surrounded by a frame. the outside edges of the frame are x inches in length and the frame is 1 inch thick. what is the total area of the frame?
A: -4x + 4**??**
B: 4x - 4
C: x^2 + 4x + 4
D: x^2 - 4x - 4
area of square outside frame = x^2
area of painting inside frame = (x-2)^2
so
x^2 - (x^2 -4 x + 4)
= 4 x -4
thank you ^^
To find the total area of the frame, we need to subtract the area of the square painting from the area of the frame including the painting.
Let's assume the side length of the square painting is s inches. Since the frame is 1 inch thick, the length of the outer square (including the frame) will be s + 2 inches on each side.
Therefore, the area of the frame including the painting is (s+2)^2 square inches.
The area of the square painting is s^2 square inches.
Subtracting the area of the painting from the total area of the frame, we get:
Total area of the frame = (s+2)^2 - s^2
Expanding the expression, we have:
Total area of the frame = s^2 + 4s + 4 - s^2
Simplifying:
Total area of the frame = 4s + 4
Since we are given that the outside edges of the frame are x inches in length, we can equate s = x.
Total area of the frame = 4x + 4
Therefore, the answer is:
A: -4x + 4 (Incorrect)
B: 4x - 4 (Incorrect)
C: x^2 + 4x + 4 (Incorrect)
D: x^2 - 4x - 4 (Incorrect)
The correct answer is missing from the given options. The correct answer should be 4x + 4.
To find the total area of the frame, we need to determine the area of the outer rectangle formed by the frame and subtract the area of the inner square painting.
Let's start by finding the length and width of the outer rectangle. Since the frame is 1 inch thick on all sides, we can add 2 inches to both the length and the width of the square painting to account for the frame.
The length of the outer rectangle is x inches (from the given information) plus 2 inches for the frame on each side, resulting in x + 2 + 2 = x + 4 inches.
The width of the outer rectangle is also x inches (from the given information) plus 2 inches for the frame on each side, resulting in x + 2 + 2 = x + 4 inches.
Therefore, the length and width of the outer rectangle are both x + 4 inches.
The area of the outer rectangle is found by multiplying the length by the width: (x + 4) * (x + 4) = (x + 4)^2.
Now, let's find the area of the inner square painting. The length of the square painting is x inches, and the width is also x inches. Therefore, its area is x * x = x^2.
To find the area of the frame, we subtract the area of the inner square painting from the area of the outer rectangle:
Area of the frame = Area of the outer rectangle - Area of the inner square painting
= (x + 4)^2 - x^2
= x^2 + 8x + 16 - x^2
= 8x + 16
So, the correct answer is option A: -4x + 4.