A square painting is surrounded by a 3-cm-wide frame. If the total area of the painting plus the frame is 81 cm2, find the dimensions of the painting.
Why did the painting go to the gym?
Because it wanted to work on its frame!
Let's start by assuming the side length of the square painting is x cm.
Since the frame is 3 cm wide on each side, the actual dimensions of the painting (excluding the frame) will be (x - 6) cm.
The area of the frame is given by (x^2 - (x - 6)^2) cm^2.
The total area of the painting plus the frame is 81 cm^2, so we can set up the equation:
x^2 - (x - 6)^2 + (x^2 - (x - 6)^2) = 81.
Now let's solve the equation step by step:
1. Distribute the squared term:
x^2 - (x^2 - 12x + 36) + (x^2 - (x^2 - 12x + 36)) = 81.
2. Simplify by combining like terms:
x^2 - x^2 + 12x - 36 + x^2 - x^2 + 12x - 36 = 81.
3. Combine like terms to get:
24x - 72 = 81.
4. Add 72 to both sides of the equation:
24x = 153.
5. Divide both sides by 24 to solve for x:
x = 6.375.
So, the side length of the square painting is approximately 6.375 cm.
To find the actual dimensions of the painting, we subtract 6 cm (twice the 3 cm frame width) from each side:
Actual dimensions = (x - 6) cm = (6.375 - 6) cm = 0.375 cm.
Therefore, the square painting has dimensions of approximately 0.375 cm x 0.375 cm.
To find the dimensions of the painting, we need to solve this problem step by step. First, let's define the dimensions of the painting.
Let's say the length and width of the painting are 'x' cm each.
Now, let's calculate the dimensions of the frame. The frame surrounds the painting with a width of 3 cm on all sides. So, the length and width of the frame can be calculated as follows:
Length of frame = Length of painting + 2 * Width of frame = x + 2 * 3
Width of frame = Width of painting + 2 * Width of frame = x + 2 * 3
Now, we can calculate the total area of the painting plus the frame:
Total Area = (Length of painting + Length of frame) * (Width of painting + Width of frame)
Given that the total area is 81 cm², we can set up the equation:
81 = (x + 2 * 3) * (x + 2 * 3)
Simplifying the equation, we get:
81 = (x + 6) * (x + 6)
81 = (x + 6)²
Taking the square root of both sides, we get:
√81 = x + 6
9 = x + 6
Simplifying further, we find:
x = 9 - 6
x = 3
So, the length and width of the painting are both 3 cm.
Therefore, the dimensions of the painting are 3 cm by 3 cm.
Let painting height/width be x.
(x+6)^2 = 81
x + 6 = 9
x = 3