Respuesta :
Answer:
(a) 162.5 ft by 325 ft.
(b)[tex]A(w)=650w-2w^{2}[/tex]
(c)Width = 162.5 feet
(d)[tex]52812.5 ft^{2}[/tex]
Step-by-step explanation:
Let w be the width of the enclosure (perpendicular to the barn)
Let l be the length of the enclosure (parallel to the barn).
Perimeter of the fence = 650 feet
Since one side will be against the barn and we are fencing only three sides,
Perimeter = 2w+l
Therefore:
2w+l =650
- l=650-2w
Area of the pen A(l,w)=lw
Substituting l=650-2w into A(l,w)
[tex]A(w)= w(650-2w)=650w-2w^{2}[/tex]
The dimension that will maximize the area of the pen occurs when the derivative of A(w)=0. (i.e. at the maximum point)
[tex]A^{'}(w)=650-4w[/tex]
650-4w=0
4w=650
w=162.5 feet
Recall: l=650-2w
l=650 -2(162.5)=650-325=325 feet
l=325 feet
(a)The dimensions that will maximize the area of the pen is 162.5 ft by 325 ft.
(b)
If w =the width of the enclosure
l = the length of the enclosure.
Area of the enclosure = lw
[tex]A(w)=650w-2w^{2}[/tex]
(c)Width, w that would maximize the area = 162.5 feet
(d)Maximum Area
[tex]A(w)=650(162.5)-2(162.5)^{2}\\=105625-52812.5\\=52812.5 ft^2[/tex]
