Sheila has a picture frame that measures 20 cm by 30 cm. If she puts a mat inside the frame that decrease the picture by 264cm^2, how wide is the mat?
A1 = L*W =20 * 30 = 600cm^2.
Am = 264cm^2 = Area of mat.
Am = 20W = 264,
20W = 264,
W = 264 / 20 = 13.2cm.
OR
30W = 264,
W = 264 / 30 = 8.8CM.
To find the width of the mat, we need to find the area of the picture frame and then subtract the area of the picture itself.
The area of the picture frame can be found by multiplying the length and width:
Area of picture frame = 20 cm × 30 cm = 600 cm^2
Let's assume the width of the mat is x cm. The dimensions of the picture will then be reduced by 2x (one x for each side).
The dimensions of the picture, after the mat is inserted, will be:
Length of picture = 20 cm - (2x)
Width of picture = 30 cm - (2x)
The area of the picture can be calculated by multiplying these dimensions:
Area of picture = (20 cm - 2x) × (30 cm - 2x)
The problem states that the area of the mat is 264 cm^2, so we can set up an equation:
600 cm^2 - 264 cm^2 = (20 cm - 2x) × (30 cm - 2x)
Simplifying the equation, we get:
336 cm^2 = (20 cm - 2x) × (30 cm - 2x)
Expanding the equation:
336 cm^2 = (600 cm^2 - 40 cm x - 60 cm x + 4x^2)
Combining like terms:
336 cm^2 = 600 cm^2 - 100 cm x + 4x^2
Moving all terms to one side:
0 = 600 cm^2 - 336 cm^2 - 100 cm x + 4x^2
Combining like terms again:
0 = 264 cm^2 - 100 cm x + 4x^2
To solve this quadratic equation, we can either factor it or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
where a = 4, b = -100, and c = 264.
Plugging in the values:
x = (-(-100) ± √((-100)^2 - 4(4)(264))) / (2(4))
x = (100 ± √(10000 - 4224)) / 8
x = (100 ± √5776) / 8
x = (100 ± 76) / 8
Simplifying further:
x1 = (100 + 76) / 8 = 176 / 8 = 22 cm
x2 = (100 - 76) / 8 = 24 / 8 = 3 cm
The width of the mat can be either 3 cm or 22 cm.
To find the width of the mat, we need to determine the decrease in the area of the picture caused by the mat.
The area of the picture frame without the mat can be calculated by multiplying the length (30 cm) by the width (20 cm).
Area of the picture frame = length * width
= 30 cm * 20 cm
= 600 cm^2
Now, let's assume the width of the mat as 'x' cm. The new dimensions of the picture inside the mat would be (30 cm - 2x) by (20 cm - 2x) because the mat would reduce both the length and width of the picture frame by twice its width.
The area of the picture inside the mat can be calculated by multiplying the new length (30 cm - 2x) by the new width (20 cm - 2x).
Area of the picture inside the mat = (30 cm - 2x) * (20 cm - 2x)
According to the given problem, the area of the picture inside the mat decreases by 264 cm^2 compared to the area of the picture frame.
So we can set up the equation:
600 cm^2 - 264 cm^2 = (30 cm - 2x) * (20 cm - 2x)
336 cm^2 = (30 cm - 2x) * (20 cm - 2x)
Now we can solve this equation to find the value of 'x', which represents the width of the mat.