A painting measuring 100cm by 55cm is to be framed with a matte border of uniform width surrounding the painting. the combined area of the picture and the matte is 9796cm^2. How wide is the matte border?
let x = the width of the border
The combined area of the picture with the border = (100+x)(55+x)
(100 + x)(55 + x) = 9796
5500 + 155x + x^2 = 9796
-4296 + 155x + x^2 = 0
Now, use the quadratic equation to find x.
x = (-b +/- sqrt(b^2 - 4ac))/2a
With these numbers, I assume a calculator is allowed.
x = 24, -179
The width of the border cannot be negative, so x = 24cm.
To find the width of the matte border, we need to set up an equation using the given information.
Let's assume the width of the matte border is "x" cm.
The dimensions of the painting along with the matte border will be:
Length: 100cm + 2x
Width: 55cm + 2x
The area of the painting can be calculated by multiplying the length and width:
Area of the painting = (100cm + 2x) * (55cm + 2x)
According to the problem, the combined area of the picture and the matte border is given as 9796cm^2. So, we can set up the following equation:
(100cm + 2x) * (55cm + 2x) = 9796cm^2
Now we can solve this equation to find the value of "x", which represents the width of the matte border.
1. Expand the equation:
(100cm + 2x) * (55cm + 2x) = 9796cm^2
5500cm^2 + 100cm * 2x + 55cm * 2x + 4x^2 = 9796cm^2
2. Simplify the equation:
5500cm^2 + 200cmx + 110cmx + 4x^2 = 9796cm^2
4x^2 + 310cmx + 5500cm^2 - 9796cm^2 = 0
4x^2 + 310cmx - 4296cm^2 = 0
3. Rearrange the equation in the form of quadratic equation:
4x^2 + 310cmx - 4296cm^2 = 0
4. Solve the quadratic equation:
Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a), where a = 4, b = 310cm, and c = -4296cm^2.
Substituting the values in the quadratic formula:
x = (-310cm ± √((310cm)^2 - 4 * 4 * -4296cm^2)) / (2 * 4)
x = (-310cm ± √(96100cm^2 + 68736cm^2)) / 8
x = (-310cm ± √(164836cm^2)) / 8
x = (-310cm ± 406cm) / 8
5. Simplify the equation:
x = (96cm / 8) or x = (-716cm / 8)
Considering the width cannot be negative, the solution would be:
x = 12cm
Therefore, the width of the matte border is 12cm.