Given:
The sum of the base and height of a triangle is 12cm.
To find:
The largest possible area of the triangle.
Explanation:
According to the problem,
[tex]\begin{gathered} x+h=12........(1) \\ x=12-h........(2) \end{gathered}[/tex]
Using the area formula of the triangle,
[tex]\begin{gathered} A=\frac{1}{2}bh \\ A=\frac{1}{2}x\cdot h.........(3) \end{gathered}[/tex]
Substituting equation (2) in (3),
[tex]\begin{gathered} A(h)=\frac{1}{2}(12-h)h \\ A(h)=6h-\frac{h^2}{2}..........(4) \end{gathered}[/tex]
Using the first derivative test,
[tex]\begin{gathered} A^{\prime}(h)=0 \\ 6-\frac{2h}{2}=0 \\ 6-h=0 \\ -h=-6 \\ h=6 \end{gathered}[/tex]
Since,
[tex]A^{\prime}^{\prime}(h)=-2<0[/tex]
So, it has a maximum value of h = 6.
Substituting h = 6 in equation (2) we get,
[tex]\begin{gathered} x=12-6 \\ x=6 \end{gathered}[/tex]
Therefore, the largest possible area will be,
[tex]\begin{gathered} A=\frac{1}{2}bh \\ =\frac{1}{2}\times6\times6 \\ =18cm^2 \end{gathered}[/tex]
Final answer:
The largest possible area of the triangle is 18 square cm.
