Let the length of the porch be \( l \) feet and the width be \( w \) feet. We have two equations based on the given information:
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The perimeter of the porch is given by: \[ 2l + 2w = 84 \] Dividing the entire equation by 2, we can simplify it to: \[ l + w = 42 \]
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The area of the porch is given by: \[ lw = 392 \]
Now, we can express \( w \) in terms of \( l \) from the first equation: \[ w = 42 - l \]
Next, we substitute \( w \) into the area equation: \[ l(42 - l) = 392 \] This expands to: \[ 42l - l^2 = 392 \] Rearranging it gives us a quadratic equation: \[ l^2 - 42l + 392 = 0 \]
We can solve this quadratic equation using the quadratic formula, where \( a = 1 \), \( b = -42 \), and \( c = 392 \): \[ l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Calculating the discriminant: \[ b^2 - 4ac = (-42)^2 - 4(1)(392) = 1764 - 1568 = 196 \] Now, substituting the values into the formula: \[ l = \frac{42 \pm \sqrt{196}}{2(1)} = \frac{42 \pm 14}{2} \] This results in two possible solutions for \( l \): 1. \[ l = \frac{42 + 14}{2} = \frac{56}{2} = 28 \] 2. \[ l = \frac{42 - 14}{2} = \frac{28}{2} = 14 \]
Now, we find the corresponding widths \( w \):
- If \( l = 28 \): \[ w = 42 - 28 = 14 \]
- If \( l = 14 \): \[ w = 42 - 14 = 28 \]
Thus, the dimensions of the porch are: \[ \text{Dimensions: } 28 \text{ feet by } 14 \text{ feet} \]
In conclusion, the dimensions of the front porch are 28 feet by 14 feet.