Question

If a rancher plans to fence a rectangular pasture adjacent to a river. The rancher has 100 meters of fencing, and no fencing is needed along the river.

How do you write an area of A as a function of x? Then I would have to find domain afterwards

Answer 1

he perimeter of the area is the length of the fence. Perimeter = x + y + y. This can be simplified to Perimeter = x + 2y. You're told that the rancher has 100 m of fence, so the perimeter is 100 m: 100 = x + 2y (1) The area of a rectangle is given by the formula A = length * width. The length in this case is x, and the width is y. So, A = xy. Solve equation (1) for y. Once you find the expression to which y equals, plug that for y into A = xy. 100 = x + 2y. Solve for y by subtracting both sides by x: 100 - x = 2y. Divide both sides by 2: (100 - x)/2 = y Substitute (100 - x)/2 for y into A = xy: A = x[(100 - x)/2] A = [x(100 - x)]/2 You can rewrite this as (100x - x²)/2 or 50x - (x²/2) or 50x - (1/2x²).

Answer 2

The perimeter of the area is the length of the fence. Perimeter = x + y + y. This can be simplified to Perimeter = x + 2y. You're told that the rancher has 100 m of fence, so the perimeter is 100 m: 100 = x + 2y (1) The area of a rectangle is given by the formula A = length * width. The length in this case is x, and the width is y. So, A = xy. Solve equation (1) for y. Once you find the expression to which y equals, plug that for y into A = xy. 100 = x + 2y. Solve for y by subtracting both sides by x: 100 - x = 2y. Divide both sides by 2: (100 - x)/2 = y Substitute (100 - x)/2 for y into A = xy: A = x[(100 - x)/2] A = [x(100 - x)]/2 You can rewrite this as (100x - x²)/2 or 50x - (x²/2) or 50x - (1/2x²). Hope that helped!