Answer:
The maximum height of the prism is [tex]12\ m[/tex]
Step-by-step explanation:
Let
x------> the height of the prism
we know that
the area of the rectangular base of the prism is equal to
[tex]A=L*W[/tex]
[tex]A\leq 27\ m^{2}[/tex]
so
[tex]L*W\leq 27[/tex] -------> inequality A
[tex]W=x-9[/tex] ------> equation B
[tex]L=W+6[/tex] -----> equation C
Substitute equation B in equation C
[tex]L=(x-9)+6[/tex]
[tex]L=x-3[/tex] ------> equation D
Substitute equation B and equation D in the inequality A
[tex](x-3)*(x-9)\leq 27[/tex]-------> using a graphing tool to solve the inequality
The solution for x is the interval---------->[tex][0,12][/tex]
see the attached figure
but remember that
The width of the base must be [tex]9[/tex] meters less than the height of the prism
so
the solution for x is the interval ------> [tex](9,12][/tex]
The maximum height of the prism is [tex]12\ m[/tex]
