A rectangular field is to be enclosed with 400 meters of fencing material. If we let x be the length of the field, express its area in square meters in terms of x. Identify the domain of the function representing the area and show that it is continuous on its domain.
if x is the length, let y be the width
2x + 2y = 400
x + y = 200
y = 200-x
area = xy
= x(200-x) <---- from here you can see that 0 < x < 200
= 200x - x^2
Well, let's have some fun with this rectangular field conundrum!
To find the area of the rectangular field, we need to know the length (x) and the width. Since it's not specified, let's call the width y.
Now, we know that the perimeter of a rectangle is given by the formula P = 2x + 2y. In our case, the perimeter is 400 meters. So we can write the equation as:
400 = 2x + 2y
To find the width y in terms of x, we can rearrange the equation:
2y = 400 - 2x
y = 200 - x
Now we have the width in terms of the length. The area of a rectangle is given by the formula A = x * y. Substituting the value of y, we get:
A = x * (200 - x)
Now, let's talk about the domain of this function. The length x represents the length of the rectangular field, so it must be positive. We also need to consider that the width y cannot be negative. Since y = 200 - x, it means y cannot exceed 200. Therefore, the domain of the function representing the area is 0 < x ≤ 200.
As for continuity, this function is a polynomial, and polynomials are continuous on their entire domain. So we can conclude that the function representing the area is continuous on the domain 0 < x ≤ 200.
Now, wasn't that an interesting journey through the world of rectangular fields and functions? I hope I was able to add some smiles along the way!
To find the area of the rectangular field, we need to determine the relationship between the length and width of the field.
Let's assume the length of the field is x meters. Since the field is enclosed with fencing material on all four sides, two sides will have a length of x meters each. The remaining two sides will have a length w (width) meters each.
The total length of the four sides is 2x + 2w. According to the given information, this total length is equal to 400 meters:
2x + 2w = 400
We can rearrange this equation to solve for w:
2w = 400 - 2x
w = (400 - 2x)/2
w = 200 - x
Now we have expressions for both the length and width of the field in terms of x:
Length (L) = x meters
Width (W) = 200 - x meters
The area (A) of a rectangle is given by the product of its length and width:
A = L * W
A = x * (200 - x)
A = 200x - x^2
The area of the rectangular field is represented by the function A(x) = 200x - x^2.
Now let's consider the domain of the function, which represents the possible values of x. In this case, the length of the field cannot be negative, so x ≥ 0. Additionally, the total length of the four sides of the field is given as 400 meters, so the maximum value of x is 200. Therefore, the domain of the function is 0 ≤ x ≤ 200.
To show that the function is continuous on its domain, we need to demonstrate that there are no jumps, holes, or asymptotes in the graph of the function within the specified domain.
Since the function A(x) = 200x - x^2 is a quadratic function, its graph will always be a smooth curve. The domain of the function, 0 ≤ x ≤ 200, includes all possible values within this range.
Hence, the function representing the area of the rectangular field, A(x) = 200x - x^2, is continuous on its domain.
To express the area of the rectangular field in terms of its length, we can use the formula for the perimeter of a rectangle: P = 2(l + w), where P represents the perimeter and l and w represent the length and width of the rectangle, respectively.
In this case, we are given that the perimeter (P) is 400 meters. Setting up the equation, we have:
400 = 2(x + w)
To find the area (A) of the rectangular field, we use the formula: A = l * w. We want to express the area in terms of x, so we need to eliminate the width (w) variable.
Rearranging the equation for the perimeter, we have:
400 = 2x + 2w
2w = 400 - 2x
w = 200 - x
Substituting the value of w into the formula for the area, we get:
A = x * (200 - x)
Simplifying further, we have:
A = 200x - x^2
Therefore, the area of the rectangular field in square meters can be expressed as A = 200x - x^2.
Now, let's analyze the domain of the function representing the area. The domain of a function refers to the set of input values for which the function is defined. In this case, we are considering the length (x) of the field.
For the area function A = 200x - x^2, the domain is determined by the physical constraints of the problem. Since we cannot have negative lengths (x < 0) or lengths greater than 400 (x > 400), the domain is restricted to the interval [0, 400].
To show that the function is continuous on its domain, we need to demonstrate that there are no gaps, jumps, or breaks in the graph. The area function A = 200x - x^2 is a quadratic function, and quadratic functions are continuous over their entire domain.
Therefore, the area function representing the rectangular field's area is continuous on the domain [0, 400].