Let the length of the mirror be \( l \) feet and the width be \( w \) feet. We have the following two equations based on the area and perimeter of the rectangle:
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The area equation: \[ l \times w = 20 \]
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The perimeter equation: \[ 2l + 2w = 18 \quad \text{or simplified,} \quad l + w = 9 \]
Now, we can solve these two equations. From the second equation, we can express \( l \) in terms of \( w \):
\[ l = 9 - w \]
Now, we substitute this expression for \( l \) into the area equation:
\[ (9 - w) \times w = 20 \]
Expanding this gives:
\[ 9w - w^2 = 20 \]
Rearranging the equation, we get a quadratic equation:
\[ w^2 - 9w + 20 = 0 \]
To solve this quadratic equation, we can factor it:
\[ (w - 4)(w - 5) = 0 \]
Setting each factor to zero gives us the possible values for \( w \):
\[ w - 4 = 0 \quad \Rightarrow \quad w = 4 \] \[ w - 5 = 0 \quad \Rightarrow \quad w = 5 \]
Now, substituting these values back to find \( l \):
If \( w = 4 \): \[ l = 9 - 4 = 5 \]
If \( w = 5 \): \[ l = 9 - 5 = 4 \]
Thus, the dimensions of the mirror are \( 4 \) feet by \( 5 \) feet.
So the answer is:
4 feet by 5 feet.
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