Respuesta :
Answer:
[tex]310\text{ feet and }210\text{ feet}[/tex]
Step-by-step explanation:
GIVEN: A farmer has [tex]520 \text{ feet}[/tex] of fencing to construct a rectangular pen up against the straight side of a barn, using the barn for one side of the pen. The length of the barn is [tex]310 \text{ feet}[/tex].
TO FIND: Determine the dimensions of the rectangle of maximum area that can be enclosed under these conditions.
SOLUTION:
Let the length of rectangle be [tex]x[/tex] and [tex]y[/tex]
perimeter of rectangular pen [tex]=2(x+y)=520\text{ feet}[/tex]
[tex]x+y=260[/tex]
[tex]y=260-x[/tex]
area of rectangular pen [tex]=\text{length}\times\text{width}[/tex]
[tex]=xy[/tex]
putting value of [tex]y[/tex]
[tex]=x(260-x)[/tex]
[tex]=260x-x^2[/tex]
to maximize [tex]\frac{d \text{(area)}}{dx}=0[/tex]
[tex]260-2x=0[/tex]
[tex]x=130\text{ feet}[/tex]
[tex]y=390\text{ feet}[/tex]
but the dimensions must be lesser or equal to than that of barn.
therefore maximum length rectangular pen [tex]=310\text{ feet}[/tex]
width of rectangular pen [tex]=210\text{ feet}[/tex]
Maximum area of rectangular pen [tex]=310\times210=65100\text{ feet}^2[/tex]
Hence maximum area of rectangular pen is [tex]65100\text{ feet}^2[/tex] and dimensions are [tex]310\text{ feet and }210\text{ feet}[/tex]
