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Picture frame has its length 8cm longer than its width. It has an inner 1-cm boundary
such that a maximum 660cm^2- picture may fit into it. Find the dimension of this frame.
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The problem formulation as it is given in the post is INCORRECT.
The correct formulation, in my version, is THIS
Picture frame has its length 8cm longer than its width. It has an inner 1-cm boundary
such that the area of the picture itself is 660cm^2. Find the dimension of this frame.
Below is my solution for this edited formulation.
Since the width of the frame is uniform and since the outer length of the frame is 8 cm longer
than its outer width, the length of the picture itself, without considering the frame, is
8 cm longer than the picture width.
So, if x is the picture width, in centimeters, then the picture length is (x+8) cm,
and we have this equation
x*(x+8) = 660 cm^2. (1)
Now you can guess mentally the solution to this equation: it is x= 22 cm for the picture width,
giving 22+8 = 30 cm for the picture length.
After this guessing, note that the left side of the equation (1) is the monotonic function,
so the guessed solution is UNIQUE.
Alternatively, you can solve this quadratic equation (1) formally, using factoring or the quadratic formula.
In either case, you will get the same solution for x.
Thus the picture is 22 x 30 centimeters.
It implies that the outer dimensions of the frame are 24 x 32 cm.
ANSWER. The outer dimensions of the frame are 24 x 32 cm.
Solved.