The owner of the Rancho Grande has 3028 yd of fencing with which to enclose a rectangular piece of grazing land situated along the straight portion of a river. If fencing is not required along the river, what are the dimensions of the largest area he can enclose?
shorter side yd
longer side yd
What is this area?
Well, I heard the owner of the Rancho Grande wants to enclose a rectangular piece of land along a river. But remember, fences aren't needed along the river because rivers already have great borders - no need to mess with what Mother Nature has already fenced off, right?
Anyway, to figure out the dimensions of the largest area, we can do some fancy math. Let's call the shorter side of the rectangular land "x" yards. Since we don't need to fence along the river, the longer side will be 3028 - 2x yards.
Now, to find the area, we just multiply the longer side by the shorter side: Area = x * (3028 - 2x).
But here's the catch - we want to maximize the area. So let's find the value of x that will make the area as big as possible!
Okay, okay, enough with the math lesson. Let's get to the answer. The dimensions of the largest area will have a shorter side of yd and a longer side of yd. And the area of this magnificent land will be square yards.
Now, go forth and enjoy your grazing land! But don't forget to invite me to your next cow-themed party. I'll bring the rubber noses and water-squirting flowers!
To find the dimensions of the largest area that can be enclosed with the given amount of fencing, we can use the concept of optimization.
Let's assume the shorter side of the rectangular grazing land is y yards.
Since the fencing is not required along the river, we only need to enclose three sides of the rectangular area. Therefore, the amount of fencing required can be calculated as:
Perimeter = 2(shorter side) + longer side
3028 = 2y + longer side
Now, let's solve this equation for the longer side:
longer side = 3028 - 2y
The area of a rectangle can be calculated as the product of its length and width. In this case, the length is the longer side and the width is the shorter side. Therefore, the area (A) can be calculated as:
A = longer side * shorter side
A = (3028 - 2y) * y
A = 3028y - 2y^2
Now, to find the maximum area, we need to find the value of y that maximizes the area. We can do this by taking the derivative of the area equation and setting it equal to zero, then solving for y:
dA/dy = 3028 - 4y
Setting dA/dy = 0:
3028 - 4y = 0
4y = 3028
y = 757
So, the shorter side of the rectangular grazing land should be 757 yards.
To find the longer side, substitute this value of y back into the equation for the longer side:
longer side = 3028 - 2y
longer side = 3028 - 2(757)
longer side = 3028 - 1514
longer side = 1514
Therefore, the dimensions of the largest area that can be enclosed with the given amount of fencing are:
Shorter side: 757 yards
Longer side: 1514 yards
The area of this rectangle can be calculated as:
Area = shorter side * longer side
Area = 757 * 1514
Area = 1,146,298 square yards
To find the dimensions of the largest area that can be enclosed, we need to find the dimensions that will maximize the area.
Let's assume the shorter side of the rectangular grazing land is x yards.
Since there is no fencing required along the river, the length of the rectangular grazing land will be twice the length of the shorter side. So, the longer side of the rectangular grazing land will be 2x yards.
To find the perimeter of the fencing, we add up all the sides of the rectangular grazing land, excluding the side along the river:
Perimeter = x + x + 2x = 4x
Given that the owner has 3028 yards of fencing, the perimeter should be equal to 3028 yards:
4x = 3028
To find the value of x, we can divide both sides of the equation by 4:
x = 3028 / 4
x = 757 yards
The shorter side of the rectangular grazing land is 757 yards, and the longer side is twice that, or 2 * 757 = 1514 yards.
The area of a rectangle is given by the formula: area = length * width.
So, the area of the rectangular grazing land is:
Area = 757 * 1514 = 1,144,798 square yards.
Therefore, the dimensions of the largest area that can be enclosed are:
Shorter side: 757 yards
Longer side: 1514 yards
And the area is: 1,144,798 square yards.