1000m of fencing is used to make a rectangular enclosure. Find the greatest possible area and the corresponding dimensions.
1000m of fencing is used to make a rectangular enclosure. Find the greatest possible area and the corresponding dimensions.
the rectangle with the greatest area is a square ... four equal sides
To find the rectangular enclosure with the greatest possible area using 1000m of fencing, we can use the concept of optimizing a function.
Let's denote the length of the enclosure as "l" and the width as "w".
The perimeter of a rectangle is given by the formula: perimeter = 2l + 2w.
In this case, we know that the perimeter is 1000m, so we can write the equation:
2l + 2w = 1000
We want to find the maximum possible area, which is given by the formula: area = length × width.
From the equation for the perimeter, we can solve it for one variable and substitute it into the formula for the area.
1. Solve the equation 2l + 2w = 1000 for one variable:
2l = 1000 - 2w
l = (1000 - 2w) / 2
l = 500 - w
2. Substitute l = 500 - w into the formula for the area:
area = (500 - w) × w
area = 500w - w^2
Now, we have the area as a function of one variable, w. To find the maximum area, we can take the derivative of the area function and set it equal to zero.
3. Calculate the derivative of the area function with respect to w:
area' = 500 - 2w
4. Set the derivative equal to zero and solve for w:
500 - 2w = 0
2w = 500
w = 250
Now, we have found the value of w where the area is maximized. By substituting this value back into the equation for l, we can find the corresponding length.
5. Substitute w = 250 into l = 500 - w:
l = 500 - 250
l = 250
So, the corresponding dimensions for the greatest possible area are length = 250m and width = 250m.
To find the maximum area, we can substitute these values into the area function:
area = 500w - w^2
area = 500(250) - (250)^2
area = 125000 - 62500
area = 62500m^2
Therefore, the greatest possible area for the rectangular enclosure is 62500 square meters, with dimensions of 250m by 250m.
To find the greatest possible area of a rectangular enclosure using 1000m of fencing, we can use the fact that the perimeter of a rectangle is equal to the sum of all its sides.
Let's assume the length of the rectangle is 'l' and the width is 'w'.
The perimeter of the rectangle is given by: 2l + 2w. Since we have a total of 1000m of fencing available, we can write the equation:
2l + 2w = 1000
We can rearrange this equation to express one variable in terms of the other. Let's solve for 'l':
2l = 1000 - 2w
l = (1000 - 2w) / 2
l = 500 - w
Now, we can express the area of the rectangle in terms of 'l' and 'w'. The area is given by the product of the length and width:
Area = l * w
= (500 - w) * w
= 500w - w^2
To find the greatest possible area, we need to maximize this equation. We can do this by finding the maximum point of the quadratic equation, which corresponds to its vertex. The vertex of the quadratic equation in the form ax^2 + bx + c is given by x = -b / (2a).
In our case, the equation is -w^2 + 500w. So, the x-coordinate of the vertex is:
w = -(-500) / (2 * -1)
w = 500 / 2
w = 250
Now, we can substitute this value of 'w' back into the equation for 'l' to find the corresponding length:
l = 500 - w
l = 500 - 250
l = 250
Therefore, the greatest possible area of the rectangular enclosure is achieved when the dimensions are 250m by 250m. The corresponding area is:
Area = l * w
= 250 * 250
= 62,500 square meters