To find the dimensions of the poster, we can set up a system of equations based on the area and the perimeter.
Let:
- \( l \) = length of the poster
- \( w \) = width of the poster
From the problem, we have the following information:
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Area: \[ l \cdot w = 10 \]
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Perimeter: \[ 2l + 2w = 14 \] Dividing the perimeter equation by 2 gives: \[ l + w = 7 \]
Now we can solve this system of equations. From the second equation, we can express \( l \) in terms of \( w \): \[ l = 7 - w \]
Now substitute \( l \) into the area equation: \[ (7 - w) w = 10 \]
Expanding this gives: \[ 7w - w^2 = 10 \] Rearranging into a standard quadratic equation: \[ w^2 - 7w + 10 = 0 \]
Next, we can factor or use the quadratic formula to solve for \( w \). The equation factors nicely: \[ (w - 5)(w - 2) = 0 \]
This gives us: \[ w = 5 \quad \text{or} \quad w = 2 \]
Using \( w = 5 \): \[ l = 7 - 5 = 2 \]
Using \( w = 2 \): \[ l = 7 - 2 = 5 \]
Thus, the dimensions of the poster are: \[ \boxed{5} \text{ feet by } \boxed{2} \text{ feet} \]