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 150yd of fencing is available, find the largest total area that can be enclosed.

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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  150yd of fencing is available, find the largest total area that can be enclosed.

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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 * 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) * 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 = -b/(2a). In our case, a = -2 and b = 150, so the y-coordinate of the vertex is y = -150/(2*(-2)) = 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 * y = 75 * 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.

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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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 150yd 150 \mathrm{yd} 150yd of fencing is available, find the largest total area that can be enclosed.

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.