A rectangular plot requires 2000 m of fencing to enclose it. If one of the dimensions is x meters, express the area y (m2) as a function of x. Determine the range of x.
Sehr
To determine the area y as a function of x, let's break down the problem step-by-step.
Step 1: Understand the problem.
We have a rectangular plot that requires 2000 m of fencing to enclose it. One of the dimensions is x meters.
Step 2: Identify the known information.
- Perimeter of a rectangle = 2 * length + 2 * width
- Perimeter of the plot = 2000 m
- One dimension of the plot is x meters.
Step 3: Determine the dimensions using the perimeter equation.
Since the perimeter is the sum of all sides, we can write the equation:
2 * length + 2 * width = 2000
Dividing both sides by 2 gives us:
length + width = 1000
Since one of the dimensions is x, we can rewrite the equation as:
x + width = 1000
width = 1000 - x
Step 4: Determine the area equation.
The area of a rectangle is given by the equation:
Area = length * width
Substituting the width we found in Step 3, we have:
y = x * (1000 - x)
Therefore, the area y is expressed as a function of x:
y = 1000x - x^2
Step 5: Determine the range of x.
To determine the range of x, we need to consider the domain of the function. In this case, since the width of a rectangle cannot be negative, we must ensure that x is a non-negative value.
We also need to consider that the length of a rectangle should always be positive to ensure a valid plot.
Therefore, the range of x is:
0 ≤ x ≤ 1000
In summary, the area y is expressed as a function of x: y = 1000x - x^2, and the range of x is 0 ≤ x ≤ 1000.
To express the area y as a function of x, let's first consider the perimeter of the rectangular plot. The given information tells us that the plot requires 2000 m of fencing to enclose it.
The perimeter of a rectangle is calculated by adding the lengths of all its sides. In this case, since we know one side has a length of x meters, let's assume that this side is the width of the rectangular plot.
Let's denote the length of the rectangle as L and the width as W. Thus, we have:
2L + 2W = 2000
Since one of the dimensions is x meters, we can substitute W with x in the equation:
2L + 2x = 2000
Now, let's solve this equation for L:
2L = 2000 - 2x
L = (2000 - 2x) / 2
L = 1000 - x
The area of a rectangle is calculated by multiplying its length by its width. So, we can express the area y as a function of x:
y = L * W
y = (1000 - x) * x
y = 1000x - x^2
Now that we have the equation for the area y as a function of x, let's determine the range of x.
Since the length and width of a rectangle cannot be negative, we can conclude that x cannot be negative in this case. Therefore, the range of x is x ≥ 0 (x is greater than or equal to 0).
the other dimension is
(2000-2x)/2 = 1000-x
I assume you know how to express the area of a rectangle.