A rancher needs to enclose two adjacent rectangular corrals, one for cattle and one for sheep. If the river forms one side of the corrals and $150 \mathrm{yd}$ of fencing is available, find the largest total area that can be enclosed.

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Q. A rancher needs to enclose two adjacent rectangular corrals, one for cattle and one for sheep. If the river forms one side of the corrals and

150 \mathrm{yd}

of fencing is available, find the largest total area that can be enclosed.

Perimeter Equation: We need to maximize the area of two adjacent rectangles with a fixed perimeter. The total perimeter available for the three sides not adjacent to the river is $150$ yards. Let's denote the length of the side parallel to the river as $x$ and the two identical sides perpendicular to the river as $y$ . The total perimeter is then $2y + x = 150$ .
Area Calculation: The total area $A$ to be maximized is the sum of the areas of the two rectangles, which is $A = x \times y$ . Since we have two rectangles, we need to account for the shared side along the river, so we only count it once.
Express Area in Terms of y: To express the area in terms of one variable, we can solve the perimeter equation for $x$ : $x = 150 - 2y$ .
Find Quadratic Function: Substitute $x$ in the area equation with the expression we found in terms of $y$ : $A = (150 - 2y) \cdot y$ .
Find Vertex: Now we have the area as a function of $y$ : $A(y) = 150y - 2y^2$ . To find the maximum area, we need to find the vertex of this quadratic function, since the coefficient of $y^2$ is negative, indicating a downward opening parabola.
Find x Value: The vertex of a parabola in the form of $A(y) = ay^2 + by + c$ is at $y = -\frac{b}{2a}$ . In our case, $a = -2$ and $b = 150$ , so the y-coordinate of the vertex is $y = -\frac{150}{2*(-2)} = \frac{150}{4} = 37.5$ .
Calculate Maximum Area: Substitute $y = 37.5$ back into the perimeter equation to find $x$ : $x = 150 - 2(37.5) = 150 - 75 = 75$ .
Final Result: Now we can find the maximum area using the values of $x$ and $y$ : $A = x \times y = 75 \times 37.5 = 2812.5$ square yards.
Final Result: Now we can find the maximum area using the values of $x$ and $y$ : $A = x \times y = 75 \times 37.5 = 2812.5$ square yards.We have found the maximum area that can be enclosed with $150$ yards of fencing for two adjacent rectangular corrals with one side formed by the river.