- Define the height as \(x\), the width as \(x-9\), and the length as \(x-3\).
- Establish the inequality based on the base area constraint: \((x-9)(x-3) \leq 27\).
- Simplify the quadratic inequality to \(x^2 - 12x \leq 0\), which factors to \(x(x-12) \leq 0\).
- Determine the valid range \(9 < x \leq 12\) by considering the physical constraints of the prism.
- Identify the maximum height: \(\boxed{12}\) meters.
Explanation
- Analyzing the Problem Constraints
Let \(x\) represent the height of the rectangular prism in meters.
Given the problem constraints:
- The width of the base is \(w = x - 9\).
- The length of the base is \(l = w + 6 = (x - 9) + 6 = x - 3\).
- For the physical dimensions to be valid, both the width and the length must be positive:
- \(w = x - 9 > 0 \implies x > 9\)
- \(l = x - 3 > 0 \implies x > 3\)
- The base area is constrained to be no more than \(27\) square meters:
- \(Area = w \times l = (x - 9)(x - 3) \leq 27\)
- Setting Up and Simplifying the Inequality
Expanding the base area expression:
\((x - 9)(x - 3) = x^2 - 3x - 9x + 27 = x^2 - 12x + 27\)
Substituting this into the inequality:
\(x^2 - 12x + 27 \leq 27\)
Subtracting \(27\) from both sides of the inequality:
\(x^2 - 12x \leq 0\)
Factoring the quadratic expression:
\(x(x - 12) \leq 0\)
- Solving for the Maximum Height
The quadratic inequality \(x(x - 12) \leq 0\) is satisfied when \(x\) is between the roots \(0\) and \(12\), inclusive, because the parabola opens upwards.
This gives the interval:
\(0 \leq x \leq 12\)
However, we must also satisfy the physical constraint \(x > 9\). Combining these two conditions:
\(9 < x \leq 12\)
Thus, the possible values for the height \(x\) lie within the interval \((9, 12]\). The maximum height of the prism is therefore \(12\) meters.
Examples
This problem models real-world design constraints, such as ensuring a container fits within a specific footprint while maintaining a required height. Engineers and architects often use these types of algebraic inequalities to determine the maximum dimensions for manufacturing parts or planning room layouts to meet strict building codes and space limitations. By calculating the upper bound of a dimension based on area, one can ensure structural stability and compliance with design specifications.