A rectangular pen for a pet is 8 feet longer than it is wide. Give possible values for the width, W, of the pen if its area must be greater than 209 square feet.
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w(w+8) > 209
(w-11)(w+19) > 0
w > 11
To find the possible values for the width of the pen, we need to consider that the area of the rectangular pen must be greater than 209 square feet.
Let's proceed step by step:
1. Let's assume the width of the pen is W in feet.
2. According to the given information, the length of the pen is 8 feet longer than its width, so the length would be (W + 8) feet.
3. The area of a rectangle is calculated by multiplying the width by the length: Area = Width * Length.
4. We know that the area must be greater than 209 square feet, so we can write the inequality: Width * Length > 209.
5. Substituting the values, Width * (W + 8) > 209.
6. Expanding the multiplication gives us W^2 + 8W > 209.
7. Rewriting the inequality in standard form, we get W^2 + 8W - 209 > 0.
8. To solve this quadratic inequality, we can find the values of W that satisfy the equation W^2 + 8W - 209 = 0.
9. We can factor or use the quadratic formula to find the solutions for W.
Factoring: (W - 11)(W + 19) = 0. The factors are W - 11 = 0, which gives W = 11, and W + 19 = 0, which gives W = -19. Since width cannot be negative, we disregard W = -19.
Quadratic formula: W = (-8 ± √(8^2 - 4 * 1 * (-209))) / (2 * 1). Simplifying gives W ≈ 11.12 and W ≈ -19.12. Again, we disregard the negative value.
10. Therefore, the possible value for the width, W, is approximately 11 feet.
So, the possible value for the width of the pen would be around 11 feet if the area of the pen has to be greater than 209 square feet.