A photograph measuring 8 inches by 6 inches is surrounded by a mat. The mat has the same width on all sides of the photograph. The photograph and mat are put into a glass photo frame which just covers the outside of the mat.
If the area of the glass surface is 120 square inches, whats the width of the mat?
hi internet
To find the width of the mat, we can start by determining the total width and height of the photo frame.
Let's assume that the width of the mat is "x" inches.
The photograph itself measures 8 inches by 6 inches. Since the mat has the same width on all sides, the dimensions of the picture including the mat would be 8 + 2x inches by 6 + 2x inches.
Now, let's calculate the total width and height of the photo frame:
Width of the frame = Width of the picture + 2 * width of the mat
= (8 + 2x) + 2(x)
= 8 + 4x
Height of the frame = Height of the picture + 2 * width of the mat
= (6 + 2x) + 2(x)
= 6 + 4x
Since the frame covers the outside of the mat, the dimensions of the frame should be larger than the mat.
The area of the glass surface is given as 120 square inches. The formula for the area of a rectangle is A = length * width. Here, the length is the width of the frame and the width is the height of the frame:
A = length * width
120 = (8 + 4x) * (6 + 4x)
To solve this quadratic equation, we can either expand and simplify the equation or use factoring.
Expanding the equation:
120 = 48 + 44x + 16x + 4x^2
4x^2 + 60x + 72 = 0
Factoring the quadratic equation:
(4x + 6)(x + 12) = 0
Setting each factor equal to zero and solving for x:
4x + 6 = 0 or x + 12 = 0
Solving for x:
4x = -6 or x = -12
Since the width cannot be negative, the solution is x = -12 is extraneous.
Thus, the width of the mat is x = -6/4 = -3/2 inches.
However, since width cannot be negative, the result is not a valid solution. In this case, it seems that there may be an error or inconsistency in the given information or the problem itself.
To have 120 in.^2 = 12 by 10
12 - 8 = ?
10 - 6 = ?