a rectangular field with an area of 8000 m^2 is enclosed by 400 m of fencing , detemine the dimesions of the field to the nearest tenth of a metre
first we represent the unknowns using variables:
let x = length
let y = width
recall that perimeter of rectangle is just,
P = 2x + 2y
since P is given (which is 400), we can get an expression of width in terms of length:
400 = 2x + 2y
400 - 2x = 2y
y = 200 - x
then to get the dimensions, we use the area given. recall that area of rectangle is just,
A = xy
substituting,
8000 = x(200 - x)
8000 = 200x - x^2
x^2 - 200x + 8000 = 0
then solve for x using quadratic equation. then also solve for y.
hope this helps~ :)
To determine the dimensions of the rectangular field, we can use the given information about the area and perimeter.
Let's assume the length of the field is L and the width is W.
1. The area of a rectangle is given by the formula A = L * W. In this case, the area is given as 8000 m^2. So we have:
8000 = L * W ----(Equation 1)
2. The perimeter of a rectangle is given by the formula P = 2L + 2W. In this case, the perimeter is given as 400 m. So we have:
400 = 2L + 2W ----(Equation 2)
To solve this system of equations, we can isolate one variable in terms of the other and substitute it into the other equation.
Let's solve Equation 2 for L:
400 = 2L + 2W
400 - 2W = 2L
200 - W = L ----(Equation 3)
Now substitute Equation 3 into Equation 1:
8000 = L * W
8000 = (200 - W) * W
8000 = 200W - W^2
Rearrange the equation into a quadratic form:
W^2 - 200W + 8000 = 0
Now, we can solve this quadratic equation to find the values of W.
Using the quadratic formula, W = [-b ± sqrt(b^2 - 4ac)] / (2a), where a, b, and c are the coefficients of the quadratic equation.
In this case, a = 1, b = -200, and c = 8000.
W = [-(-200) ± sqrt((-200)^2 - 4 * 1 * 8000)] / (2 * 1)
W = [200 ± sqrt(40000 - 32000)] / 2
W = [200 ± sqrt(8000)] / 2
W = [200 ± 89.44] / 2
We have two solutions for W:
1. W = (200 + 89.44) / 2 = 144.72 m
2. W = (200 - 89.44) / 2 = 55.56 m
Substituting these values into Equation 3, we can find the corresponding values for L:
1. L = 200 - W = 200 - 144.72 = 55.28 m
2. L = 200 - W = 200 - 55.56 = 144.44 m
So, the dimensions of the rectangular field to the nearest tenth of a meter are approximately 55.3 m by 144.4 m.
Well, let's solve this problem with a touch of humor, shall we?
To find the dimensions of the rectangular field, we'll need to do some math. But don't worry, numbers and formulas aren't as scary as a clown car!
Let's call the length of the field "L" and the width "W". The perimeter of a rectangle is equal to twice the length plus twice the width.
So we have the equation: 2L + 2W = 400 meters of fencing.
But we also know that the area of the field is 8000 square meters. The area of a rectangle is simply the product of its length and width.
So we have another equation: L * W = 8000 square meters.
Now, let's solve this mystery! We can rewrite the first equation as L + W = 200.
Now, we have a system of equations to solve simultaneously. But don't worry, we're not trying to solve a riddle here!
Let's use the substitution method. We can solve the first equation for L, getting L = 200 - W.
Now we can substitute this value of L into the second equation: (200 - W) * W = 8000.
Simplifying, we get the quadratic equation W^2 - 200W + 8000 = 0.
Using the quadratic formula or a calculator, we find that W ≈ 44.7 meters (to the nearest tenth).
Now we can substitute this value of W back into the first equation to find L: L ≈ 200 - 44.7 ≈ 155.3 meters (to the nearest tenth).
So, the dimensions of the field are approximately 155.3 meters by 44.7 meters.
And there you have it! The dimensions of the field to the nearest tenth of a meter. Good luck with your rectangular clown adventures!
To determine the dimensions of the rectangular field, we can start by setting up two equations based on the given information.
Let's assume the length of the field to be L and the width to be W.
Equation 1: The Area of the rectangular field = Length × Width
L × W = 8000
Equation 2: The Perimeter of the rectangular field = 2 × Length + 2 × Width
2L + 2W = 400
Now, we can solve these two equations simultaneously to find the values of L and W.
Step 1: Rearrange Equation 2 to solve for one variable in terms of the other:
2L = 400 - 2W
L = 200 - W
Step 2: Substitute the value of L from Step 1 into Equation 1:
(200 - W) × W = 8000
Step 3: Simplify the equation:
200W - W^2 = 8000
Step 4: Rearrange the equation and set it to zero:
W^2 - 200W + 8000 = 0
Step 5: Solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, we can factor the equation:
(W - 80)(W - 120) = 0
This yields two possible solutions for W: W = 80 or W = 120.
Step 6: Substitute the values of W back into Equation 2 to find L:
When W = 80, L = 200 - 80 = 120
When W = 120, L = 200 - 120 = 80
Therefore, the dimensions of the rectangular field are approximately 120 meters by 80 meters or 80 meters by 120 meters.