Let's carefully analyze the given function and identify the true statement based on that analysis.
The function that models the area [tex]\( G(x) \)[/tex] of the rectangular garden, where [tex]\( x \)[/tex] is the width, is given by:
[tex]\[ G(x) = -2x^2 + 36x \][/tex]
To find the [tex]\( x \)[/tex]-intercepts, we need to solve the equation:
[tex]\[ -2x^2 + 36x = 0 \][/tex]
To solve this equation, we can factor out the common term [tex]\( x \)[/tex]:
[tex]\[ x(-2x + 36) = 0 \][/tex]
Setting each factor to zero gives us:
[tex]\[ x = 0 \quad \text{or} \quad -2x + 36 = 0 \][/tex]
Solving the second equation for [tex]\( x \)[/tex]:
[tex]\[ -2x + 36 = 0 \][/tex]
[tex]\[ -2x = -36 \][/tex]
[tex]\[ x = 18 \][/tex]
Therefore, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = 0 \)[/tex] and [tex]\( x = 18 \)[/tex].
The [tex]\( x \)[/tex]-intercepts represent the points where the function [tex]\( G(x) \)[/tex] intersects the x-axis, which in this context means the possible values of [tex]\( x \)[/tex] (the width of the garden) where the area is zero. These intercepts also serve as the lower and upper bounds for the possible lengths of the garden.
Thus, the correct statement is:
The [tex]\( x \)[/tex]-intercepts of the function are 0 and 18 and are the lower and upper bounds for the possible lengths of the garden.
In Arabic:
نقاط تقاطع الدالة مع المحور الأفقي [tex]\( x \)[/tex] يكون عند 0 و 18 وهذا أصغر وأكبر قيمتان للطول الممكن للحديقة.
