To find the dimensions of the airplane ticket, we can start by letting the length of the ticket be \( l \) centimeters and the width be \( w \) centimeters. We know the following two equations based on the perimeter and area provided:
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The perimeter \( P \) of a rectangle is given by the formula: \[ P = 2l + 2w \] Given that the perimeter is 36 centimeters, we have: \[ 2l + 2w = 36 \] Simplifying this gives: \[ l + w = 18 \quad \text{(Equation 1)} \]
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The area \( A \) of a rectangle is given by the formula: \[ A = lw \] Given that the area is 80 square centimeters, we have: \[ lw = 80 \quad \text{(Equation 2)} \]
Now we can use Equation 1 to express one variable in terms of the other. From Equation 1: \[ w = 18 - l \]
Substituting this into Equation 2: \[ l(18 - l) = 80 \] Expanding this, we get: \[ 18l - l^2 = 80 \] Rearranging the equation gives us a standard quadratic equation: \[ l^2 - 18l + 80 = 0 \]
Next, we can apply the quadratic formula \( l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -18 \), and \( c = 80 \): \[ l = \frac{18 \pm \sqrt{(-18)^2 - 4 \cdot 1 \cdot 80}}{2 \cdot 1} \] Calculating the discriminant: \[ b^2 - 4ac = 324 - 320 = 4 \] Thus, \[ l = \frac{18 \pm \sqrt{4}}{2} \] Calculating the square root: \[ l = \frac{18 \pm 2}{2} \] This gives us two values for \( l \): \[ l = \frac{20}{2} = 10 \quad \text{and} \quad l = \frac{16}{2} = 8 \]
So, the dimensions of the ticket can be:
- \( l = 10 \) cm and \( w = 18 - 10 = 8 \) cm
- \( l = 8 \) cm and \( w = 18 - 8 = 10 \) cm
This means the dimensions of the ticket are: \[ \text{10 centimeters by 8 centimeters} \] or equivalently, \[ \text{8 centimeters by 10 centimeters.} \]
Thus, the final answer is: 10 centimeters by 8 centimeters.