A matte of uniform width is to be placed around a painting so that the area of the matted surface is equal to the area of the painting. If the dimensions of the painting are 15 cm and 10 cm, find the width of the matte. The answer is supposed to be 2.5 cm but I keep getting 12.5.
subtract the area of the painting from the total area. If the matte has width w, then
(10+2w)(15+2w) - 10*15 = 10*15
4w^2 + 50w - 150 = 0
2(2w-5)(w+15) = 0
w = 5/2 = 2.5
To find the width of the matte that needs to be placed around the painting, we can set up an equation using the areas of the matted surface and the painting.
Let's denote the width of the matte as "x" centimeters.
The area of the painting is given by the product of its length and width: 15 cm * 10 cm = 150 cm².
The area of the matted surface can be found by adding 2 times the width of the matte to the dimensions of the painting and then multiplying the resulting length and width. So, the area of the matted surface is:
(15 cm + 2x) * (10 cm + 2x)
Now, since it's mentioned that the area of the matted surface is equal to the area of the painting, we can set up the following equation:
(15 cm + 2x) * (10 cm + 2x) = 150 cm²
Expanding this equation:
(150 cm² + 30 cmx + 20 cmx + 4x²) = 150 cm²
Combining like terms:
4x² + 50 cmx + 150 cm² = 150 cm²
Subtracting 150 cm² from both sides:
4x² + 50 cmx = 0
Now, we can solve this quadratic equation by factoring it:
x(4x + 50 cm) = 0
This equation will be true when either x = 0 (which is not a feasible solution in this context) or when 4x + 50 cm = 0.
Solving the equation 4x + 50 cm = 0 for x:
4x = -50 cm
x = -50 cm / 4
x = -12.5 cm
So, it seems that there was an error in your calculation. The correct value for the width of the matte is x = -12.5 cm, not x = 2.5 cm. Please double-check your calculations and make sure you haven't made any mistakes.
To find the width of the matte, we can solve this problem algebraically.
Let's assume the width of the matte is 'x' cm.
The dimensions of the matted surface will be:
Length: (15 + 2x) cm
Width: (10 + 2x) cm
Now, we need to set up an equation to find the width of the matte.
The area of the matted surface = Area of the painting
(15 + 2x) * (10 + 2x) = 15 * 10
Expanding the equation, we get:
150 + 20x + 30x + 4x^2 = 150
Simplifying the equation, we get:
4x^2 + 50x + 150 - 150 = 0
4x^2 + 50x = 0
Dividing the equation by 2, we get:
2x^2 + 25x = 0
Factoring out x, we get:
x(2x + 25) = 0
From the equation, we have two possible solutions:
1) x = 0
2) 2x + 25 = 0
Since the width cannot be zero (it must be greater than zero for the matte to exist), we can disregard the solution x = 0.
Solving the equation 2x + 25 = 0, we get:
2x = -25
x = -25/2
x = -12.5
Here's where the error occurs. The width of the matte cannot be negative since it represents a physical measurement.
Therefore, there must be a mistake in the calculation or in setting up the equation. Please double-check the calculations and ensure that the equation has been set up correctly.
The correct answer is indeed x = 2.5 cm, as the width of the matte should be a positive value.