a rancher has 310 feet of fencing with which to enclose two rectangular corrals, both of the same size. the two corrals will share one side, and a barn forms one side of both corrals. suppose the width of each corral is X feet. express the total area of the two corrals as a function of X.
assuming the barn forms the "length" of each corral, then the total fencing used comprises three widths and two lengths.
Letting x be the width and y the length,
3x + 2y = 310
y = (310-3x)/2
the area of each corral is width * length, so the total area is 2xy
area = 2xy = 2x * (310-3x)/2 = x(310-3x)
To find the total area of the two corrals, we need to determine the dimensions of each corral and then calculate their individual areas.
Let's break down the problem step by step:
1. Start by identifying the dimensions of one corral. Since the width of each corral is X feet, let's denote the width of one corral as W1.
2. Now, let's consider the length of one corral. The total fencing available is 310 feet, and we know that two sides of each corral will be formed by fencing. Therefore, the length of each corral will be (310 - 2X) feet.
3. To find the area of each corral, multiply the length and width: A1 = W1 * (310 - 2X).
4. Since the two corrals are of the same size, the area of the second corral will be the same as the first corral: A2 = W1 * (310 - 2X).
5. Lastly, to express the total area of the two corrals as a function of X, we add the areas of the two corrals: Total Area = A1 + A2 = 2 * A1 = 2 * (W1 * (310 - 2X)) = 2W1 * (310 - 2X).
Therefore, the total area of the two corrals can be expressed as a function of X: Total Area = 2W1 * (310 - 2X).