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