a picture 8 in. by 12 in. is placed in a frame which has uniform width. if the area of the frame equals the area of the picture, find the dimension of the frame
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picture area = 12x8 = 96
total area (picture+frame) is
(8+2w)(12+2w) = 96+96
w = 2
now just get the frame dimensions, knowing width=2.
Well, well, well, let's solve this picture-perfect puzzle! So, we have a picture that is 8 inches by 12 inches. Now, let's assume the uniform width of the frame as "x" inches.
To find the area of the frame, we need to subtract the area of the picture from the total area of the frame and the picture combined. The total area of the frame and picture combined can be found by increasing the dimensions of the picture by twice the width of the frame.
So, the area of the frame would be (8 + 2x) * (12 + 2x) square inches, and the area of the picture is 8 * 12 square inches.
Now here comes the funny bit: since the area of the frame equals the area of the picture, we can write an equation:
(8 + 2x) * (12 + 2x) = 8 * 12
Expanding this equation, we get:
96 + 32x + 16x + 4x^2 = 96
Simplifying it further:
4x^2 + 48x = 0
Dividing by 4, we get:
x^2 + 12x = 0
Factoring out an "x," we have:
x(x + 12) = 0
So we get two possible solutions: x = 0 or x = -12. However, we can't have a negative width for the frame, so the only possible solution is x = 0.
Now, don't panic, my friend! If x = 0, it means there's no frame at all. So, in this particular scenario, the dimension of the frame is zero inches. Aren't you glad to have a frameless picture? It's light and airy!
Let's assume the width of the frame is "x".
The dimension of the frame on the top and bottom would be:
Width = 8 in + 2x
The dimension of the frame on the sides would be:
Length = 12 in + 2x
The area of the picture is given by multiplying the length and width of the picture:
Area of the picture = 8 in * 12 in = 96 in^2
The area of the frame can be calculated by subtracting the area of the picture from the area of the frame including the picture:
Area of the frame = (Width) * (Length) - Area of the picture
Area of the frame = (8 in + 2x) * (12 in + 2x) - 96 in^2
Since the area of the frame equals the area of the picture:
Area of the frame = 96 in^2
We can set up the equation:
(8 in + 2x) * (12 in + 2x) - 96 in^2 = 96 in^2
Simplifying the equation:
(8 in + 2x) * (12 in + 2x) = 192 in^2
Expanding the equation:
96 in^2 + 16 inx + 24 inx + 4x^2 = 192 in^2
Combining like terms:
120 inx + 4x^2 = 96 in^2
Rearranging the equation:
4x^2 + 120 inx - 96 in^2 = 0
Dividing the equation by 4:
x^2 + 30 inx - 24 in^2 = 0
Unfortunately, this equation does not factor nicely. We will use the quadratic formula to solve it:
x = (-30 ± sqrt(30^2 - 4*1*(-24)))/(2*1)
x = (-30 ± sqrt(900 + 96))/(2)
x = (-30 ± sqrt(996))/(2)
x ≈ (-30 ± 31.57)/(2)
The two possible solutions for x are:
x ≈ (-30 + 31.57)/2 = 1.57/2 = 0.785 in
x ≈ (-30 - 31.57)/2 = -61.57/2 = -30.785 in
Since the dimension of the frame cannot be negative, the width of the frame is approximately 0.785 inches or 0.79 inches (rounded to two decimal places).
To find the dimensions of the frame, we need to determine the width of the frame.
Let's assume the width of the frame is "x".
Since the picture and the frame have the same area, we can set up the following equation:
Area of the picture = Area of the frame
The area of the picture can be calculated by multiplying its length by its width:
8 in. * 12 in. = 96 in²
The area of the frame is the total area minus the area of the picture. Since the frame is uniform all the way around, we have:
Area of the frame = (length of frame + 2 * width of frame) * (width of frame + 2 * width of frame) - Area of the picture
Substituting the given measurements:
Area of the frame = (8 in. + 2x) * (12 in. + 2x) - 96 in²
Now, we need to set up the equation, as the area of the frame equals the area of the picture:
(8 in. + 2x) * (12 in. + 2x) - 96 in² = 96 in²
Expanding and simplifying the equation:
(96 + 32x + 4x²) - 96 = 0
This simplifies to:
4x² + 32x = 0
Now, we can factor out 4x:
4x(x + 8) = 0
Setting each factor equal to 0:
4x = 0 or x + 8 = 0
This gives us two possible solutions:
1. 4x = 0
x = 0
2. x + 8 = 0
x = -8
Since we cannot have a negative width for the frame, we discard the second solution.
Therefore, the width of the frame (x) is 0.
Since the width is 0, the frame does not have any width or thickness.