Final answer:
To determine the minimum width of the pen, one must solve the quadratic inequality created by setting the width as 'w' and the length as 'w+4' feet. After expanding and factoring the inequality, the positive solution is chosen as the width, since it cannot be negative. The calculation results in a minimum width of 2 feet for the pen.
Step-by-step explanation:
The student has asked for the minimum width of a rectangular pen where the length is four feet more than the width and the area must be at least 12 square feet. Let us denote the width of the pen as 'w' feet. Considering the description, the length would be 'w + 4' feet. Using the formula for the area of a rectangle (Area = length × width), we can set up the inequality:
w × (w + 4) ≥ 12
This is a quadratic inequality. To solve it, first, we expand the equation:
w^2 + 4w ≥ 12
Subtract 12 from both sides to get the quadratic equation:
w^2 + 4w - 12 ≤ 0
Next, we factor the quadratic equation (if possible) or use the quadratic formula to find the values of 'w' that satisfy the inequality. Once we find the possible values, we must choose the one that gives the width, which can't be negative, so we take the positive solution. This will be the minimum width needed for the pen.
Example of Possible Calculation Steps:
- Factor the quadratic equation: (w + 6)(w - 2) ≤ 0
- Find the 'w' values where the inequality is satisfied: w = -6 or w = 2
- Since width cannot be negative, w = 2 feet is the minimum width of the pen.