Solution to 7. Separation of variables become dr ue integral evaluate to: In 3y +2 = + c Apply e to get: 3y + 2 ca. Some algebra yields the general solution: y implies c4 = 26 So the particular solution is: | y 26e3a- -dr. Use 1-substitution with u 3y+2 and - dy. The integrals 1 3y + 2 l du Apply some algebra to get: In 3y +2 = 3x + c2 2 c4e3 Use IC to find es: 8-y(0) Exponential Growth and Mixing problems homework to be handed in. Just two problems: Problem 1. (Exponential growth) Suppose a small colony of 1,000 bacteria, at t 0 minutes is in a large bottle of water with lots of food and no predators (so there are no predation or density dependent issues to worry minutes there are 2,500 bacteria in the colony. Let y(t) = the number o in a large about). Suppose after 45 f bacteria in the colony at time t, t measured in minutes Write the ODE IVP for the above scenario and find the solution to the IVP. Use the solution to estimate how many bacteria in the bottle at t 90 minutes. Circle your answer (the number of bacteria after 90 minutes). were lem 2. (Mixing) We start with 2,000 liters of seawater in a tank. In seawater there are 35 g of salt per liter. Suppose we continually pour fresh water (0.5 g salt/liter) into the top of the tank at the rate of 10 liters/minute and at the bottom of the tank, we continually drain off 10 liters/minute. So the amount of water in the tank is always 2,000 liters. Assume that the water in the tank is being stirred so that the saltwater and the freshwater mix immediately. How long until the concent of ration salt in the tank is 6 g/liter? Circle your final answer (how many minutes until the salt concentration is 6 g/L) Instructions. On page 1, immediately after your name, write: (1) Number of bacteria after 90 min = your answer (2) Minutes until the salt concentration is 6 g/Lyo Then, underneath and on the following pages show all work. Don't skip steps. Show the integration. Use a pencil (so you can erase your mistakes). Be neat so that I can follow your work. Circle final answers. STAPLE with stapler the pages. No paper clips, etc.

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