a square painting is surrounded by a 2.5cm frame. if the total are of the painting plus frame is 3600 sq cm, find the dimensions of the painting.
f = frame = 2.5 cm
p = dimension of frame
p + f = dimension of one side of square
Area of a square painting :
A = ( p + f ) ^ 2 = 3600
p + f = sqrt ( 3600 )
p + f = 60
p + 2.5 = 60
p = 60 - 2.5 = 57.5 cm
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Duplicate post. This question has been answered already
To find the dimensions of the painting, we need to subtract the area of the frame from the total area of the painting plus frame.
Let's denote the side length of the square painting as "x".
The area of the frame is equal to the difference between the area of the square with side length (x + 2.5 cm) and the area of the square with side length "x".
Area of the frame = (x + 2.5)^2 - x^2
According to the given information, the total area of the painting plus frame is 3600 sq cm. So we can set up the following equation:
Total Area = Area of the Painting + Area of the Frame
3600 = x^2 + (x + 2.5)^2 - x^2
Simplify the equation by expanding and combining like terms:
3600 = x^2 + (x^2 + 5x + 6.25) - x^2
Now, we can simplify further:
3600 = 2x^2 + 5x + 6.25
Rearrange the equation to set it equal to zero:
2x^2 + 5x + 6.25 - 3600 = 0
Combine like terms:
2x^2 + 5x - 3593.75 = 0
Now, we can solve this quadratic equation for "x" using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 2, b = 5, and c = -3593.75. Substituting these values into the formula:
x = (-5 ± √(5^2 - 4 * 2 * (-3593.75))) / (2 * 2)
Simplify further:
x = (-5 ± √(25 + 28750)) / 4
x = (-5 ± √28775) / 4
Now, we have two possible values for "x". Let's calculate both:
1. For x = (-5 + √28775) / 4, we get x ≈ 59.47
2. For x = (-5 - √28775) / 4, we get x ≈ -179.97
Since the dimensions of the painting cannot be negative, we discard the second solution.
Therefore, the side length of the square painting is approximately 59.47 cm.