To express the area A of a rectangle as a function of the width w when the width is twice the length, we start by establishing the relationship between the width and length of the rectangle.
Let the length of the rectangle be denoted as L. According to the problem, the width (W) is twice the length, which can be mathematically expressed as:
- W = 2L
The area A of a rectangle is calculated using the formula:
- A = L × W
Now, substituting the expression for width into the area formula:
- A = L × (2L)
- A = 2L²
Next, we need to express L in terms of w. Since we have W = 2L, we can rearrange this to find L:
- L = W / 2
Now, substitute this expression for L back into the area equation:
- A = 2(W / 2)²
- A = 2(W² / 4)
- A = (1/2)W²
Therefore, the area A of the rectangle as a function of the width w is:
- A(w) = (1/2)W²