Let the length of the rug be \( l \) feet and the width be \( w \) feet. We have the following equations based on the problem's constraints:
- Area: \[ lw = 12 \]
- Perimeter: \[ 2l + 2w = 16 \] Simplifying the perimeter equation gives: \[ l + w = 8 \]
Now, we have a system of equations:
\[ lw = 12 \quad (1) \] \[ l + w = 8 \quad (2) \]
From equation (2), we can express \( w \) in terms of \( l \): \[ w = 8 - l \]
Substituting this expression for \( w \) into equation (1): \[ l(8 - l) = 12 \] Expanding this gives: \[ 8l - l^2 = 12 \] Rearranging into standard quadratic form: \[ l^2 - 8l + 12 = 0 \]
Now, we can use the quadratic formula \( l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -8 \), and \( c = 12 \): \[ b^2 - 4ac = (-8)^2 - 4 \cdot 1 \cdot 12 = 64 - 48 = 16 \] Now, substituting into the quadratic formula: \[ l = \frac{8 \pm \sqrt{16}}{2 \cdot 1} = \frac{8 \pm 4}{2} \] Calculating both possible values for \( l \): \[ l = \frac{12}{2} = 6 \quad \text{and} \quad l = \frac{4}{2} = 2 \]
Thus, we have two possible values for \( l \):
- If \( l = 6 \), then \( w = 8 - 6 = 2 \).
- If \( l = 2 \), then \( w = 8 - 2 = 6 \).
Therefore, the dimensions of the rug are \( 6 \) feet by \( 2 \) feet.
Final dimensions: \[ \text{6 feet by 2 feet} \]