To find the dimensions of the rectangular sticker, we have two equations based on the provided information:
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The area of the rectangle: \[ l \times w = 66 \] where \( l \) is the length and \( w \) is the width.
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The perimeter of the rectangle: \[ 2(l + w) = 34 \] Simplifying this, we have: \[ l + w = 17 \]
Now we have a system of equations:
- \( l \times w = 66 \)
- \( l + w = 17 \)
From the second equation, we can express \( l \) in terms of \( w \): \[ l = 17 - w \]
Now, substitute this expression for \( l \) into the first equation: \[ (17 - w) \times w = 66 \]
Expanding this gives: \[ 17w - w^2 = 66 \]
Rearranging this into a standard quadratic equation form: \[ w^2 - 17w + 66 = 0 \]
Now, we need to factor this quadratic equation. We are looking for two numbers that multiply to \( 66 \) and add up to \( 17 \). The numbers \( 11 \) and \( 6 \) satisfy these conditions, as \( 11 \times 6 = 66 \) and \( 11 + 6 = 17 \).
Thus, we can factor the equation as: \[ (w - 11)(w - 6) = 0 \]
Setting each factor equal to zero gives us: \[ w - 11 = 0 \quad \text{or} \quad w - 6 = 0 \] This leads to: \[ w = 11 \quad \text{or} \quad w = 6 \]
Now, substituting back to find \( l \):
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If \( w = 11 \): \[ l = 17 - 11 = 6 \]
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If \( w = 6 \): \[ l = 17 - 6 = 11 \]
Thus, the dimensions of the sticker are \( 11 \) cm by \( 6 \) cm. We can write the final answer as:
11 centimeters by 6 centimeters.
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