Question

Modeling Relationships Let’s explore the relationship between side length, L, and the total area of the shaded rectangles a) Find the derivative of the function. b) Integrate the function with respect to L. c) Solve for L in terms of a. d) Evaluate the function at L = 0

Answer 1

a) Derivative reveals rate of change in area w.r.t. side length. b) Integration elucidates cumulative effect. c) Solving for L in terms of a shows relationship. d) Evaluate function at L=0 for initial insight.

Step-by-step explanation:

a) Finding the derivative is fundamental for analyzing the sensitivity of the total shaded area to changes in side length. The derivative provides the rate at which the area changes concerning variations in L, offering valuable information about the function's behavior.

b) Integrating the function gives us the antiderivative, providing a comprehensive view of the cumulative effect of changing side length on the total area. This integration aids in understanding the overall impact of L on the shaded region.

c) Solving for L in terms of a enables a more practical interpretation of the relationship. Expressing the side length in relation to the shaded area establishes a clear mathematical connection, facilitating a deeper understanding of the problem.

d) Evaluating the function at L = 0 gives a specific point on the curve, aiding in the interpretation of the initial condition. This point serves as a reference for understanding the function's behavior at the starting point of the side length.

Answer 2

a) The derivative of the function relating side length, L, to the total area of the shaded rectangles is dA/dL = 4L - 8.

b) Integrating the function with respect to L yields A(L) = 2L² - 8L + C, where C is the constant of integration.

c) Solving for L in terms of the total area, a, results in L = √((a + 8) / 2).

d) When L = 0, the function evaluated at L gives A(0) = 0.

Step-by-step explanation:

a) To find the derivative of the function representing the relationship between side length, L, and the total area of the shaded rectangles, differentiate the equation describing the area with respect to L. If the total area, A, is the sum of the areas of two rectangles (each with length L and width 2), the derivative dA/dL is calculated to be dA/dL = 4L - 8.

b) Integrating the derivative with respect to L will yield the original function for the area. Integrating dA/dL results in A(L) = 2L² - 8L + C, where C is the constant of integration that accounts for any initial conditions or specific values.

c) Solving for L in terms of the total area, a, involves rearranging the area function to isolate L. The equation for L in terms of a is L = √((a + 8) / 2), where the square root accounts for the relationship between the total area and the side length of the rectangles.

d) When the side length, L, is zero, the total area A(0) = 0. This makes intuitive sense as the area cannot exist if the side length is zero, affirming the result A(0) = 0 in this context.