Bot GPT 4-o mini To solve your problem, let's break it down into two parts as specified.
Part A: Write an equation for the area of the frame.
Let:
- \( w \) = width of the frame (in feet)
- Then, the length \( l \) = \( w + 10 \) (since the length is 10 feet longer than the width)
The area \( A \) of the rectangular frame can be represented as: \[ A = l \times w \] Substituting \( l \) in the area equation gives: \[ A = (w + 10) \times w \] We know the area is 961 square feet, so we can set up the equation: \[ w(w + 10) = 961 \]
Part B: Determine the dimensions of the frame.
Now, expand and rearrange the equation from Part A: \[ w^2 + 10w - 961 = 0 \]
We can solve this quadratic equation using the quadratic formula: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = 10 \), and \( c = -961 \).
Calculating the discriminant: \[ b^2 - 4ac = 10^2 - 4 \times 1 \times -961 = 100 + 3844 = 3944 \]
Now applying the quadratic formula: \[ w = \frac{-10 \pm \sqrt{3944}}{2 \times 1} \ w = \frac{-10 \pm 62.78}{2} \]
Calculating the two possible solutions for \( w \):
- \( w = \frac{-10 + 62.78}{2} \approx \frac{52.78}{2} \approx 26.39 \)
- \( w = \frac{-10 - 62.78}{2} \approx \frac{-72.78}{2} \approx -36.39 \) (not a valid solution for width)
Therefore, we take \( w \approx 26.39 \) feet.
To find the length \( l \): \[ l = w + 10 \approx 26.39 + 10 = 36.39 \text{ feet} \]
Conclusion
The dimensions of the frame are approximately:
- Width: 26.39 feet
- Length: 36.39 feet
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