At all times, the length of a rectangle is twice the width w of the rectangleas the area of the rectangle changes with respect to time t.
a. Find an equation relating A to w.

Verified step by step guidance

1. Start by recalling the formula for the area of a rectangle, which is the product of its length and width: A = length × width.
2. According to the problem, the length of the rectangle is twice the width. Therefore, express the length in terms of the width: length = 2w.
3. Substitute the expression for the length into the area formula. This gives you A = 2w × w.
4. Simplify the equation to find the relationship between the area A and the width w: A = 2w^2.
5. This equation, A = 2w^2, now relates the area of the rectangle to its width, considering the given condition that the length is twice the width.

Key Concepts

Area of a Rectangle

The area A of a rectangle is calculated using the formula A = length × width. In this case, since the length is twice the width (l = 2w), the area can be expressed as A = 2w × w = 2w². Understanding this relationship is crucial for deriving the equation that relates area to width.

Relationship Between Variables

In calculus, understanding how different variables relate to each other is essential. Here, the area A is a function of the width w, which means any change in w will affect A. This relationship is foundational for analyzing how the area changes as the width varies.

Differentiation

Differentiation is a key concept in calculus that deals with the rate of change of a function. If the problem requires finding how the area changes with respect to time, we would differentiate the area function A(w) with respect to w and then apply the chain rule to relate it to time t, allowing us to analyze the dynamics of the rectangle's area.