Calculus AB Assignment More Applications of Exponential and Logarithmic Differential Equations 1. You make a pudding for a dinner party and put it in the refrigerator at 5 P.M. ( = 0). Your refrigerator maintains a constant temperature of 40". The pudding will be ready to serve when it cools to 45. When you put the pudding in the refrigerator you measure its temperature to be 190, and when the first guest arrives at 6 P.M., you measure it again and get a temperature reading of 100. Based on Newton's Law of Cooling, when is the earliest you can serve the pudding? Give your answer to the nearest minute. 2. Suppose you throw an object from a great height, so that it reaches very nearly terminal velocity by time it hits the ground. By measuring the impact, you determine that this terminal velocity is -49 m/sec. A. Write the equation representing the velocity v() of the object at time : seconds given the initial velocity vo and the fact that acceleration due to gravity is -9.8 m/sec-. (Here, assume you're modeling the falling body with the differential equation # = g-kv, and use the resulting formula for v(1) found in the Tutorial. Of course, you can derive it if you'd like.) B. Determine the value of , the "continuous percentage growth rate" from the velocity equation, by utilizing the information given concerning the terminal velocity.Calculus AB Assignment More Applications of Exponential and Logarithmic Differential Equations C. Using the value of & you derived above, at what velocity must the object be thrown upward if you want it to reach its peak height after 3 sec? Approximate your solution to three decimal places, and justify your answer. 3. Biologists stocked a lake with 800 fish and estimated the carrying capacity of the lake to be 20000. In the first year, the number of fish doubled. Use a logistic model to model the fish population and answer the questions below. A. Find an expression for the size of the fish population P = P() at time : in years. (You may approximate the constant & to six decimal places.) B. How long will it take for the fish population to reach 10000? (Approximate to three decimal places).Calculus AB Assignment More Applications of Exponential and Logarithmic Differential Equations 4. Consider a lake of constant volume 12200 km", which at time : contains an amount y() tons of pollutant evenly distributed throughout the lake with a concentration 12200 No tons/km Assume that fresh water enters the lake at a rate of 67.1 km/yr, and that water leaves the lake at the same rate. Suppose that pollutants are added directly to the lake at a constant rate of 550 tons/yr. Among the many simplifying assumptions that must be made to model such a complicated real-world process is that the pollutants coming into the lake are instantaneously evenly distributed throughout the lake. A. Write a differential equation for y(). B. Solve the differential equation for initial condition y(0) = 200000 to get an expression for y(1). Use your solution y() to describe in practical terms what happens to the amount of pollutants in the lake as / goes from 0 to infinity