Question

A farmer wants to fence a rectangular piece of land adjacent to the north side of his horse barn. The side adjacent to the horse barn does not need a fence and the east side borders the neighbor's house, and the neighbor will split the cost of fencing this side. The cost of fencing is 20/m and his maximum budget is5000. Find the size of the land (length and width) so that its area is the largest?

Answer 1

To find the dimensions of the land that will maximize its area, let's assume that the length of the land is x meters. The width of the land can be represented as the remaining distance after subtracting the length of the barn and the shared side. By finding the vertex of the quadratic function, we can determine that the dimensions that maximize the land's area are 125 meters by 1500 meters.

Step-by-step explanation:

To find the dimensions of the land that will maximize its area, let's assume that the length of the land is x meters. The width of the land can be represented as the remaining distance after subtracting the length of the barn and the shared side. So, the width of the land is (5000 - 20x) / 2 meters. The area of the land is found by multiplying the length and width, which gives us x * ((5000 - 20x) / 2) = -10x^2 + 2500x.

To find the maximum area, we need to find the x value that maximizes the quadratic function. We can do this by finding the vertex of the parabola. The x-coordinate of the vertex is given by -b / (2a), where a = -10 and b = 2500. Plugging in these values, we get x = -2500 / (2 * -10) = 125. Substituting this x value back into the equation for the width, we find that the width of the land is (5000 - 20 * 125) / 2 = 1500 meters. Therefore, the dimensions of the land that maximize its area are 125 meters by 1500 meters.