Consider this related rates problem. A rectangle initially has dimensions 1 cm by 4 cm. Then, the rectangle grows. All sides increase in length at the rate of 2 cm/second. At what rate is the area of the rectangle increasing after 20 seconds? Plan the approach. We need an equation that connects (or "relates") the quantities of this problem: area and side length. For a rectangle, we use the formula A = 1 . w In this scenario, all three of the quantities are changing over time, so we can write A (t) = 1(t) . w(t) Next, we apply the derivative to get a new equation that connects (or "relates") the rates of change of these quantities. In this case, we use theV [ Select to find the derivative. product rule dA = l' (t) - w (t) + I'Hopital's rule extreme value theorem Finally, we plug in the In this case, we are given l' (t) = w' (t) = 2 , and we have to figure out the number values of Z and w when t = 20, using some simple arithmetic or algebra. Final Answer: At what rate is the area of the rectangle increasing after 20 seconds? 170 square cm / secIn our class, some word problems are optimization problems, and others are related rates problems. Think about the process for solving each type of problem. Three problem-solving steps are listed below. . Which is part of the problem-solving process for optimization problems? . Which is part of the problem-solving process for related rates problems? . Which is recommended for both types of problems? Use the derivative to find the critical numbers within [ Choose ] the relevant domain. Also treat the endpoints of a Optimization closed domain as critical numbers. Both optimization and related rates Related Rates After creating an equation that connects the given [ Choose ] quantities of the problem, use implicit differentiation or the chain rule to create an equation that connects rates of change. Make a rough sketch where you can record quantities [ Choose ] from the problem, both known and unknown