two triangular pens are built against a barn. four hundred forty meters of fencing are to be used for the…

two triangular pens are built against a barn. four hundred forty meters of fencing are to be used for the three sides and the diagonal dividing fence (see figure). what dimensions maximize the area of the pen? let a be the area of the pen. what is the objective function in terms of the length of the side of the pen perpendicular to the barn, x. a = (type an expression.)

two triangular pens are built against a barn. four hundred forty meters of fencing are to be used for the three sides and the diagonal dividing fence (see figure). what dimensions maximize the area of the pen? let a be the area of the pen. what is the objective function in terms of the length of the side of the pen perpendicular to the barn, x. a = (type an expression.)

Answer

Explanation:

Step1: Define variables

Let the side perpendicular to the barn be $x$ and the side parallel to the barn be $y$. The length of the diagonal is $\sqrt{x^{2}+x^{2}}=\sqrt{2}x$ (since the two - triangular pens form a rectangle - like shape with the diagonal). The total length of the fencing is $3x + y=440$, so $y = 440 - 3x$.

Step2: Find the area formula

The area of the two - triangular pens (which is equivalent to the area of a rectangle) $A=xy$. Substitute $y = 440 - 3x$ into the area formula. So $A(x)=x(440 - 3x)=440x-3x^{2}$.

Answer:

$440x - 3x^{2}$