1. The population of a bacteria culture increases at the rate of 3 times the square root of the present population.

Transcription

1 1. The population of a bacteria culture increases at the rate of 3 times the square root of the present population. A. Model the population P = P(t) of the bacteria population with a differential equation. B. Solve the differential equation that models the population P = P(t) of the bacteria population. Solution (equation) 1 Work leading to solution 1 C. Suppose the population at time t = 0 hours is Derive an equation for the population P as an explicit function of time t (in hours). Your equation should contain no undetermined constants. Equation for P(t) containing no undetermined constants 1 Work leading to solution 2 TOTAL 3 D. What s the population of the bacteria culture at the end of 10 hours? After 100 hours? (You can round down to an integer for your population solutions.) Answers 1 Any unauthorized copying, reuse, or redistribution is prohibited. 1

2 2. A chemical reaction proceeds in such a way that after the first second, the amount of a certain chemical involved in the reaction changes at a rate that s inversely proportional to the product of the mass of the chemical present (in grams) and the time elapsed since the reaction began (in seconds). A. The mass m = m(t) of this chemical is modeled with what differential equation for time t 1 second? B. The solution to the differential equation modeling the mass m of the chemical at time t seconds is m(t) = 2k lnt + C for t 1, where k and C are undetermined constants. Show how this equation is derived from your answer in part A. Work in deriving equation 2 C. Suppose that the amount of this chemical involved in the reaction is 40 grams at time t = 1 second and 30 grams at time t = 10 seconds. Find an explicit equation for the mass m of the chemical as a function of t, fort 1. Your equation should not involve any unknown constants or any calculator numbers. Explicit equation for m = m(t) 1 Work in deriving equation 2 TOTAL 3 D. According to your equation for m(t) in part C, at what time does the mass of the chemical involved in the reaction become zero? (You may use your calculator here.) Time at which mass of chemical is zero 1 Any unauthorized copying, reuse, or redistribution is prohibited. 2

3 3. The slope of a curve is equal to y divided by 4 more than x 2 at any point (x,y) on the curve. A. Find a differential equation describing this curve. B. Solve the differential equation from part A. Solution to differential equation 1 Work in deriving equation 1 C. Suppose it s known that as x goes to infinity on the curve, y goes to 1. Find the equation for the curve by using part B and determining the constant. Explain all reasoning. Equation of the curve 1 Work/reasoning 1 Any unauthorized copying, reuse, or redistribution is prohibited. 3

4 4. At high temperatures, nitrogen dioxide, NO 2, decomposes into NO and O 2.Ify(t) is the concentration of NO 2 (in moles per liter), then at K, y(t) changes according to the reaction law dy =.05y 2 for time t in seconds. dt A. Express y in terms of t and the initial concentration y -. Equation of y in terms of t and initial concentration y - 1 Work in solving differential equation 2 Work in solving for constant of integration 1 TOTAL 4 B. Assuming that the concentration of NO 2 is twice as high at t = 20 seconds as it is at 100 seconds, find the exact initial concentration of the NO 2. Reminder: Exact means no calculator numbers. Exact initial concentration y - 1 Workin solving 1 Any unauthorized copying, reuse, or redistribution is prohibited. 4

5 5. Consider the differential equation dy dx = y(1 x). A. Use the axes provided to sketch a slope field for the given differential equation at the eleven points indicated. (2 points) B. Use the slope field for the given differential equation to explain whether a solution could have the graph shown at right: (2 points) C. On your slope field from part (a), sketch the graph of the particular solution y = f(x) such that f(0) = (1 point) Any unauthorized copying, reuse, or redistribution is prohibited. 5