Rectangle

Definition

A rectangle is a four-sided polygon, or quadrilateral, where every angle is a right angle (90 degrees) and opposite sides are equal in length. This shape is fundamental in geometry and often serves as a basis for understanding related concepts such as area, perimeter, and dimensions in various applications, including solving related rates problems.

5 Must Know Facts For Your Next Test

1. In related rates problems, rectangles can be used to represent changing dimensions over time, allowing for the calculation of how those changes affect area or perimeter.
2. The relationship between the length and width of a rectangle can lead to different rates of change, which are essential to determining rates in related rates problems.
3. When solving for related rates involving rectangles, it's common to use implicit differentiation to connect the changing dimensions with their rates.
4. The formula for area ($$A = lw$$) and perimeter ($$P = 2l + 2w$$) is crucial for setting up equations when dealing with problems that involve rectangles.
5. Understanding how one dimension affects another dimension in a rectangle (like if the length increases while maintaining a constant area) is key in solving complex related rates scenarios.

Review Questions

• How does the concept of a rectangle help in understanding relationships between different quantities in related rates problems?
  • Rectangles serve as a simple geometric model to illustrate how changing one dimension affects others, such as area or perimeter. By applying formulas like area ($$A = lw$$) and perimeter ($$P = 2l + 2w$$), you can set up relationships that reveal how the rate of change of one side influences another. This is particularly useful when dimensions are not constant and help in visualizing real-world applications of calculus.
• In what ways can the properties of rectangles be used to derive equations for related rates involving area and perimeter?
  • The properties of rectangles provide essential formulas for area and perimeter that can be differentiated to find rates of change. For instance, if you know that the area must remain constant while one dimension changes, you can express this condition mathematically by relating the lengths using the area formula. By differentiating these expressions with respect to time, you can derive equations that represent how fast one side must change as another does.
• Evaluate a problem where a rectangle's length is increasing at 2 units per second while its area remains constant at 50 square units. How does this affect the width?
  • To solve this, start with the area formula $$A = lw$$, which is set at 50. When the length $$l$$ increases at 2 units per second, you can express this as $$rac{dl}{dt} = 2$$. Since the area remains constant, you differentiate: $$0 = w rac{dl}{dt} + l rac{dw}{dt}$$. Substituting in the known values gives you $$0 = w(2) + (50/w)rac{dw}{dt}$$. Rearranging leads to finding $$rac{dw}{dt}$$, showing how fast the width must decrease as the length increases while keeping the area steady.

Related terms

Area: The measure of the space inside a two-dimensional shape, calculated for a rectangle using the formula length times width.

Perimeter: The total distance around a two-dimensional shape, for a rectangle calculated by adding twice the length and twice the width.

Similar Triangles: Triangles that have the same shape but may differ in size; their corresponding angles are equal and their sides are proportional.