Question

There is 8m wire with this wire, make a rectangle with length Xm on one side.

a. express the width using x.
b. express the area of rectangle using x .
c. Determine the domain and range of f(×).​

Answer 1

a. The width of the rectangle can be expressed as W = 4 - X. b. The area of the rectangle can be expressed as A = X * (4 - X). c. The domain of f(x) is 0 <= x <= 8 and the range is 0 <= f(x) <= 16.

Step-by-step explanation:

a. To express the width of the rectangle using x, we can use the formula for the perimeter of a rectangle. The perimeter of a rectangle is equal to twice the length plus twice the width. Since we know the length is Xm and the total length of the wire is 8m, we can set up the equation: 8 = 2X + 2W, where W represents the width of the rectangle. Solving for W, we get: W = 4 - X.

b. To express the area of the rectangle using x, we can multiply the length and width of the rectangle. The area of a rectangle is equal to the length multiplied by the width. So the area of the rectangle would be: A = X * (4 - X).

c. The domain of f(x) in this case would be the possible values of x that make sense within the context of the problem. Since the length of the rectangle cannot be negative and cannot exceed the total length of the wire (8m), the domain of f(x) would be: 0 <= x <= 8.

The range of f(x) would be the possible area values of the rectangle. Since the length and width must both be non-negative, the range of f(x) would be: 0 <= f(x) <= 16.

Answer 2

To create a rectangle from an 8m wire with length x, the width is expressed as 4 - x, the area as x(4 - x), the domain of f(x) as 0 < x < 4, and the range as 0 <= f(x) <= 4 square meters.

Step-by-step explanation:

The question pertains to using a given length of wire to create a rectangular shape and then expressing dimensions and area in terms of an algebraic variable.

Part a: Express the Width Using x

Given 8 meters of wire to form a rectangle and designating the length of one side as x meters, the perimeter (P) of the rectangle equals 2 times the length (l) plus 2 times the width (w), which can be expressed as:

P = 2l + 2w

With P as 8m and l as x, we have:

8 = 2x + 2w

Thus, the width can be expressed as w = 4 - x.

Part b: Express the Area of the Rectangle Using x

The area (A) of a rectangle is the product of its length and width:

A = l × w

Substituting the expressions for l and w, we get:

A = x(4 - x)

Part c: Determine the Domain and Range of f(x)

The function f(x) represents the area of the rectangle in terms of the length x.

Since x must be positive and cannot be equal to or greater than 4 (otherwise, the width would be zero or negative), the domain of f(x) is 0 < x < 4.

The range of f(x) would be the set of possible areas, starting from 0 and reaching a maximum when x is 2 (the rectangle becomes a square), which can be calculated by evaluating f(2):

Maximum Area = f(2)

= 2(4 - 2)

= 4 square meters

Therefore, the range of f(x) is 0 <= f(x) <= 4 square meters.