To express the area A of a rectangle as a function of the width w, given that the width is twice the length, we start with some basic definitions.
Let the length of the rectangle be denoted as l and the width as w. According to the problem statement, the width is twice the length, which we can write mathematically as:
w = 2l
Now, we know that the area A of a rectangle is calculated by the formula:
A = length × width
Substituting the expression for the width in terms of length into the area formula gives us:
A = l × w
Since w = 2l, we can replace l in the equation:
A = l × (2l) = 2l^2
Now we need to express everything in terms of the width w. We know from the relationship w = 2l that:
l = w/2
Now, substituting this back into our area equation:
A = 2(w/2)^2
This simplifies to:
A = 2(1/4)w^2 = (1/2)w^2
Thus, we can now express the area of the rectangle as a function of the width:
A(w) = (1/2)w^2
In summary, if the width of the rectangle is twice its length, the area as a function of the width is (1/2)w^2.