Question

a rectangular prism must have a base with an area of no more than 27 square meters. the width of the base must be 9 meters less than the height of the prism. the length of the base must be 6 meters more than the width of the base. find the maximum height of the prism. let x = the height of the prism x - 9 = the width of the base complete = the length of the prism x - 6 x + 6 x - 3

Explanation:

Step1: Define Variables

Let \( x \) be the height of the prism. The width of the base is \( x - 9 \) (given). The length of the base is 6 meters more than the width, so length \( = (x - 9) + 6 = x - 3 \)? Wait, no, wait. Wait, the length is 6 more than the width. Width is \( x - 9 \), so length \( = (x - 9) + 6 = x - 3 \)? Wait, no, the options are \( x - 6 \), \( x + 6 \), \( x - 3 \). Wait, let's re-express. Width \( w = x - 9 \). Length \( l = w + 6 = (x - 9) + 6 = x - 3 \)? Wait, no, that's not matching. Wait, maybe I made a mistake. Wait, the problem says "the length of the base must be 6 meters more than the width of the base". So if width is \( x - 9 \), then length \( l = (x - 9) + 6 = x - 3 \)? But the options include \( x - 3 \), \( x + 6 \), \( x - 6 \). Wait, maybe I misread. Wait, the width is 9 meters less than the height, so \( w = x - 9 \). Length is 6 more than width, so \( l = w + 6 = (x - 9) + 6 = x - 3 \). So the length is \( x - 3 \)? Wait, no, let's check again. Wait, height is \( x \), width is \( x - 9 \), length is width + 6, so \( (x - 9) + 6 = x - 3 \). So the length is \( x - 3 \). But the options have \( x - 3 \) as one of them. Wait, but maybe I messed up. Wait, maybe the width is \( x - 9 \), length is width + 6, so \( (x - 9) + 6 = x - 3 \). So the length is \( x - 3 \). So the correct option for length is \( x - 3 \)? Wait, no, the options are \( x - 6 \), \( x + 6 \), \( x - 3 \). Wait, let's re-express. Wait, width is \( x - 9 \), length is width + 6, so \( l = (x - 9) + 6 = x - 3 \). So the length is \( x - 3 \). So the answer for the length dropdown is \( x - 3 \). But maybe I made a mistake. Wait, let's check again. The problem says: "the length of the base must be 6 meters more than the width of the base". Width is \( x - 9 \), so length is \( (x - 9) + 6 = x - 3 \). So the length is \( x - 3 \). So the correct option is \( x - 3 \).

But then, to find the maximum height, we need to use the area of the base. The base is a rectangle, so area \( A = l \times w \). We know \( A \leq 27 \). So \( l \times w \leq 27 \). We have \( l = x - 3 \), \( w = x - 9 \). So \( (x - 3)(x - 9) \leq 27 \). Let's expand that: \( x^2 - 12x + 27 \leq 27 \). Subtract 27 from both sides: \( x^2 - 12x \leq 0 \). Factor: \( x(x - 12) \leq 0 \). The solutions to \( x(x - 12) = 0 \) are \( x = 0 \) and \( x = 12 \). Since \( x \) is a height, it must be positive, and \( w = x - 9 \) must also be positive (since width can't be negative), so \( x - 9 > 0 \implies x > 9 \). So the inequality \( x(x - 12) \leq 0 \) with \( x > 9 \) gives \( 9 < x \leq 12 \). So the maximum height is 12 meters. But first, let's confirm the length. Wait, the length is \( x - 3 \), width is \( x - 9 \), so area is \( (x - 3)(x - 9) \leq 27 \). Let's check when \( x = 12 \), area is \( (12 - 3)(12 - 9) = 9 \times 3 = 27 \), which is allowed. If \( x > 12 \), say \( x = 13 \), area is \( (13 - 3)(13 - 9) = 10 \times 4 = 40 > 27 \), which is not allowed. So maximum height is 12. But first, the length: since width is \( x - 9 \), length is width + 6, so \( (x - 9) + 6 = x - 3 \). So the length is \( x - 3 \), so the correct option for length is \( x - 3 \).

Step2: Solve for Maximum Height

We have the area of the base \( A = l \times w \), where \( l = x - 3 \) and \( w = x - 9 \). The area must be no more than 27, so:
\[ (x - 3)(x - 9) \leq 27 \]
Expand the left side:
\[ x^2 - 12x + 27 \leq 27 \]
Subtract 27 from both sides:
\[ x^2 - 12x \leq 0 \]
Factor:
\[ x(x - 12) \leq 0 \]
The critical points are \( x = 0 \) and \( x = 12 \). Since \( x \) (height) and \( w = x - 9 \) (width) must be positive, \( x > 9 \). The inequality \( x(x - 12) \leq 0 \) holds when \( 0 \leq x \leq 12 \). Combining with \( x > 9 \), we get \( 9 < x \leq 12 \). Thus, the maximum height is 12 meters.

Answer:

The length of the prism is \( x - 3 \), and the maximum height of the prism is 12 meters. For the length dropdown, the correct option is \( x - 3 \). The maximum height is \(\boxed{12}\) meters.