the area of rectangular plot of land fenced all round is 2000 sq. yards and the total length of fencing 180 yards. find the length and width of plot.
Let l = length
Let w = width
It was said that the total length of fencing is 180 yards. Note that perimeter of rectangle is equal to:
P = 2l + 2w
Substituting,
180 = 2l + 2w
90 = l + w
We can have an expression for length in terms of the width:
l = 90 - w
Recall that area of rectangle is length times the width. Substituting,
A = l * w
2000 = (90 - w) * w
2000 = 90w - w^2
w^2 - 90w + 2000 = 0
This is a quadratic equation, and it is immediately factorable. You'll have two roots for w, but choose only the positive, since dimensions cannot be negative. Finally, solve for the length.
Hope this helps~ `u`
x=-40,x=-50
To find the length and width of the rectangular plot, we can set up a system of equations based on the given information.
Let's assume the length of the plot is "L" yards and the width is "W" yards.
Given:
The area of the plot is 2000 sq. yards.
The perimeter of the plot is 180 yards.
We know that the area of a rectangle is given by:
Area = Length × Width
So, we can write the first equation as:
L × W = 2000 .....(1)
We also know that the perimeter of a rectangle is given by:
Perimeter = 2 × Length + 2 × Width
So, we can write the second equation as:
2L + 2W = 180 .....(2)
Now, we have a system of equations (equation 1 and equation 2) that we can solve simultaneously to find the values of L and W.
Let's solve this equation using the substitution method:
From equation 2, we can express one variable in terms of the other:
2L + 2W = 180
2L = 180 - 2W
L = (180 - 2W) / 2
L = 90 - W/2
Substituting this value of L in equation 1:
(90 - W/2) × W = 2000
90W - W^2/2 = 2000
Multiplying by 2 to eliminate the fraction:
180W - W^2 = 4000
Rearranging the equation:
W^2 - 180W + 4000 = 0
We now have a quadratic equation. Let's solve it using factoring or the quadratic formula.
By factoring or using the quadratic formula, we find that the solutions are:
W = 40 or W = 100
If W = 40:
L = 90 - W/2 = 90 - 40/2 = 90 - 20 = 70
So, the length is 70 yards and the width is 40 yards.
If W = 100:
L = 90 - W/2 = 90 - 100/2 = 90 - 50 = 40
So, the length is 40 yards and the width is 100 yards.
Therefore, the possible lengths and widths of the rectangular plot are:
Length = 70 yards, Width = 40 yards
OR
Length = 40 yards, Width = 100 yards.
To find the length and width of the rectangular plot, we need to use two equations: one equation for the area and another equation for the perimeter (total length of fencing).
Let's assume that the length of the plot is 'L' yards and the width is 'W' yards.
The equation for the area of a rectangle is: A = L * W, where A represents the area.
The equation for the perimeter of a rectangle is: P = 2L + 2W, where P represents the perimeter.
Given that the area is 2000 sq. yards and the total length of fencing is 180 yards, we can set up the following equations:
Equation 1: A = L * W = 2000
Equation 2: P = 2L + 2W = 180
To solve these equations simultaneously, we can rearrange Equation 1 to solve for one variable in terms of the other. Since we need to find the length and width, let's solve for W:
W = 2000 / L
Now substitute this value of W into Equation 2:
2L + 2W = 180
2L + 2(2000/L) = 180
Simplify the equation:
2L + 4000/L = 180
Next, multiply the entire equation by L to eliminate the fraction:
2L^2 + 4000 = 180L
Rearrange the equation:
2L^2 - 180L + 4000 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Let's use the quadratic formula:
L = (-b ± √(b^2 - 4ac)) / 2a
For this equation, a = 2, b = -180, and c = 4000. Now, substitute these values into the formula:
L = (-(-180) ± √((-180)^2 - 4(2)(4000))) / 2(2)
L = (180 ± √(32400 - 32000)) / 4
L = (180 ± √400) / 4
L = (180 ± 20) / 4
L = 200/4 = 50 or L = 160/4 = 40
Since the length cannot be longer than the perimeter, we can discard the solution L = 50. Therefore, L = 40 yards.
Substitute this value back into Equation 1 to find the width:
W = 2000 / L
W = 2000 / 40
W = 50 yards
So, the length of the plot (L) is 40 yards and the width (W) is 50 yards.