Respuesta :
The optimal lenght of x and y are:
x = 30ft
y = 30ft
To solve this, we want to know the best lenght of x and y to get the maximum area.
Then, we know the perimeter of the pens: 120ft
The perimeter is the sum of all the sides:
[tex]P=2x+2y[/tex]Sinnce we have 2 sides of lenght x and 2 sides of length y
Also, we know how the area is calculated. Base times height:
[tex]A=x\cdot y[/tex]Now, we know the perimeter he has is 120ft. Then, we can replace P = 120ft and solve for y:
[tex]\begin{gathered} P=120ft \\ 120ft=2x+2y \\ y=\frac{120ft-2x}{2} \\ y=60ft-x \end{gathered}[/tex]Now we can replace y in the area equation:
[tex]\begin{gathered} \begin{cases}A=xy \\ y=60ft-x\end{cases} \\ A=x(60ft-x) \\ A=60ft\cdot x-x^2 \end{gathered}[/tex]Here we have a function of the area respect x.
Then, we want to find the maximum area possible. We know that the zeroes of the derivative of a function is where we can find maximums and minimums.
Then let's derivate the function of the area:
[tex]\begin{gathered} A(x)=60ft\cdot x-x^2 \\ A^{\prime}(x)=60ft-2x \end{gathered}[/tex]Now let's find the zero of the derivative:
[tex]\begin{gathered} 0=60ft-2x \\ x=\frac{-60ft}{-2x} \\ x=30ft \end{gathered}[/tex]This is the value of x that maximizes the area with a perimeter of 120ft.
Now let's find the value of y:
[tex]\begin{gathered} \begin{cases}x=30ft \\ y=60ft-x\end{cases} \\ y=60ft-30ft \\ y=30ft \end{gathered}[/tex]Then the maximum area is with a lenght of y = 30ft and x = 30ft.
This also tell us that the shape that maximizes the area of a 4 sided shape with a certain perimeter is a square.
