Respuesta :
Answer:
The maximal area will be "1093.5 square feet".
Step-by-step explanation:
Let,
Length = L feet
Breadth = b feet
Given Total fencing = 162 feet
According to the question,
[tex](2\times L)+(3\times b)=162[/tex]
[tex]2L+3B=162[/tex]
[tex]L=\frac{162-3b}{2}[/tex]
[tex]L=81-\frac{3}{2}b[/tex]
As we know,
[tex]Area=Length\times breadth[/tex]
[tex]=(81-\frac{3}{2}b)\times b[/tex]
[tex]=81b-\frac{3}{2}b^2[/tex]
Now, we required to decrease or minimize the are. So for extreme points:
[tex]\frac{dA}{db}=0[/tex]
or,
[tex]\frac{dA}{dB}=\frac{d}{db}(81-\frac{3}{2}b^2 )=0[/tex]
[tex]81-\frac{3}{2}\times 2\times b=0[/tex]
[tex]b=\frac{81}{3}[/tex]
[tex]b=27 \ feet[/tex]
Now on putting the value of b, we get
[tex]l=81-\frac{3}{2}\times 27[/tex]
[tex]=81-40.5[/tex]
[tex]=40.5 \ feet\\[/tex]
So that the dimensions will be:
⇒ 40.5 feet by 27 feet
Therefore when the dimension are above then the area will be:
= [tex]81\times 27-\frac{3}{2}\times 27\times 27[/tex]
= [tex]2187-\frac{3}{2}\times 729[/tex]
= [tex]2187-1093.5[/tex]
= [tex]1093.5 \ square \ feet[/tex]
