a rectangular field is to be enclosed with 600m of fencing. Find the dimensions of the field that will give you the greatest area. please help me.. exam tomorrow!:(
200 by 100
If you are fencing all around, then a square will give you the largest area
each side will be 150 m for an area of 22500 m^2
tyler's area is only 20000 m^2
To find the dimensions of the rectangular field that will give you the greatest area, you can use the concept of calculus. The problem can be solved using optimization techniques.
Let's assume the length of the rectangular field is 'l' and the width is 'w'.
The perimeter of a rectangle can be found using the formula P = 2l + 2w.
According to the problem statement, the perimeter is 600m, so we can write the equation as:
2l + 2w = 600
Now, to find the dimensions that maximize the area, we need to find the equation for the area of a rectangle, which is A = l * w.
Using the equation for the perimeter, we can rewrite one of the variables in terms of the other. Let's solve for 'l' in terms of 'w':
2l = 600 - 2w
l = (600 - 2w)/2
l = 300 - w
Substitute this value of 'l' in the equation for the area:
A = l * w
A = (300 - w) * w
A = 300w - w^2
To find the maximum area, we need to find the maximum value of this area equation. For a quadratic equation like this, the maximum occurs when the derivative is equal to zero.
Differentiate the area equation with respect to 'w':
dA/dw = 300 - 2w
Now, set dA/dw equal to zero and solve for 'w':
300 - 2w = 0
2w = 300
w = 150
From this, we have the width 'w' equal to 150m. Plug this value back into the equation for 'l' to find the length:
l = 300 - w
l = 300 - 150
l = 150
So, the length 'l' is also 150m.
Therefore, the dimensions of the rectangular field that will give you the greatest area are length = 150m and width = 150m.
To find the dimensions of the field that will give the greatest area, we can use the method of optimization.
Let's assume the length of the rectangular field is "l" and the width of the rectangular field is "w".
According to the problem, the perimeter of the rectangular field is 600m. That means, the sum of all the four sides of the rectangle equals 600m.
Perimeter = 2(length + width)
600 = 2(l + w)
300 = l + w [Dividing both sides by 2]
Now, we need to find an equation for the area of the rectangular field in terms of one variable, either "l" or "w". Since we need to find the maximum area, it would be easier if we express the area in terms of one variable.
Area = length * width
Area = l * w
We can express one of the variables in terms of the other variable using the equation we obtained earlier for the perimeter.
w = 300 - l
Now, substitute this value of "w" in terms of "l" into the equation for the area:
Area = l * (300 - l)
Area = 300l - l^2
Now, to find the maximum area, we need to find the critical points. To do that, we take the derivative of the area equation with respect to "l" and set it equal to zero.
d(Area)/d(l) = 300 - 2l
0 = 300 - 2l
Solving this equation, we find:
2l = 300
l = 150
Substituting this value of "l" back into the equation for the width:
w = 300 - l
w = 300 - 150
w = 150
So, the dimensions of the rectangular field that will give you the greatest area are a length of 150m and a width of 150m.