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Math 2320 Differential Equations Worksheet #4

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1a) Model the growth of the population of 50,000 bacteria in a petri dish if the growth rate is k.

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1b) Suppose 3 days later, the population has grown to about 80,000 bacteria. Find the growth rate and estimate the bacteria population after 5 days. is separable. After 5 days: After 5 days, there will be approximately 109,438 bacteria in the petri dish.

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2)A fish hatchery raises trout in ponds. At the beginning of the year, the ponds contain approximately 100,000 trout. The growth rate (birthrate minus deathrate) is estimated to be about 5 per 100 per week. The hatchery wants to harvest at a constant rate of R fish per week, and increase the population by 150,000 by the end of the year. Find the appropriate harvest rate, R. Initial Conditions: Initial ODE: Solve as a Linear (or Separable) ODE: Continued on the next slide.

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2)A fish hatchery raises trout in ponds. At the beginning of the year, the ponds contain approximately 100,000 trout. The growth rate (birthrate minus deathrate) is estimated to be about 5 per 100 per week. The hatchery wants to harvest at a constant rate of R fish per week, and increase the population by 150,000 by the end of the year. Find the appropriate harvest rate, R. Initial Conditions: Initial ODE: Solve as a Linear (or Separable) ODE: Solve for R and c: Continued on the next slide.

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2)A fish hatchery raises trout in ponds. At the beginning of the year, the ponds contain approximately 100,000 trout. The growth rate (birthrate minus deathrate) is estimated to be about 5 per 100 per week. The hatchery wants to harvest at a constant rate of R fish per week, and increase the population by 150,000 by the end of the year. Find the appropriate harvest rate, R. Initial Conditions: Initial ODE: Solve as a Linear (or Separable) ODE: Solve for R and c: Trout should be harvested at a rate of approximately 4398 per week.

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3)A tank, having a capacity of 3000 gallons, initially contains 20 pounds of salt, dissolved in 1000 gallons of water. A solution containing 0.4 pounds of salt per gallon flows into the tank at a rate of 5 gallons per min and the well-stirred solution flows out of the tank at a rate of 2 gallons per minute. 3a)How much time will elapse before the tank is filled to capacity? Current capacity = Current volume + (flow in – flow out): 3000 gallons = 1000 gallons + (5 gal/min – 2 gal/min)t minutes 3000 = 1000 + 3t 2000 = 3t T = 2000/3 = 666 2/3 minutes. It will take 666 2/3 minutes for the tank to fill to capacity.

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3b) What is the salt concentration in the tank when it contains 2000 gallons of solution? Initial Condition: Initial ODE: Solve as a Linear (or Separable) ODE: Continued on the next slide.

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3b) What is the salt concentration in the tank when it contains 2000 gallons of solution? Initial Condition: Initial ODE: Solve as a Linear (or Separable) ODE: Continued on the next slide.

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3b) What is the salt concentration in the tank when it contains 2000 gallons of solution? Initial Condition: Initial ODE: Solve the ODE: Continued on the next slide. Apply the Initial Condition:

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3b) What is the salt concentration in the tank when it contains 2000 gallons of solution? Initial Condition: Initial ODE: Solve the ODE: Continued on the next slide. Apply the Initial Condition: Find the time when the tank contains 2000 gallons of solution. minutes

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3b) What is the salt concentration in the tank when it contains 2000 gallons of solution? Initial Condition: Initial ODE: Solve the ODE: Continued on the next slide. Apply the Initial Condition: Find the time when the tank contains 2000 gallons of solution. T = 1000/3 minutes When the tank contains 2000 gallons, there will be 560.62 pounds of salt in the tank. Find the salt concentration in the tank when t = 1000/3 minutes

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3c) What is the salt concentration at the instant that the tank is filled to capacity? When the tank is filled to capacity, there will be 1017.32 pounds of salt in the tank. From 3a), the tank is filled to capacity when t = 2000/3 minutes

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4)At the instant that a cake is removed from an oven, its temperature is 375 F. The cake is placed in a room whose temperature is 75 F. After 2 minutes, the cake cools to a temperature of 175 F. What is the temperature of the cake after 10 minutes? Initial Conditions: Initial ODE: Solve the separable (or linear) ODE: Continued on the next slide.

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4)At the instant that a cake is removed from an oven, its temperature is 375 F. The cake is placed in a room whose temperature is 75 F. After 2 minutes, the cake cools to a temperature of 175 F. What is the temperature of the cake after 10 minutes? Initial Conditions: Initial ODE: Solve the separable (or linear) ODE: Apply the initial conditions: Find the temperature of the cake after 10 minutes. The temperature of the cake after 10 minutes is approximately 76 degrees.

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Initial Value Problems A differential equation, together with values of the solution and its derivatives at a point x 0, is called an initial value problem.

Initial Value Problems A differential equation, together with values of the solution and its derivatives at a point x 0, is called an initial value problem.

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