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5.10: Homework- Growth and Decay

Last updated

Oct 19, 2021
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- 5.9: Growth and Decay
- 5.11: Exploring Graphs of Differential Equations

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Money that is compounded continuously follows the differential equation M′(t)=rM(t), where t is measured in years, M(t) is measured in dollars, and r is the rate. Suppose r=0.05 and M(0)=1000.
1. What is a function that satisfies this initial value problem?
  We know from class that this is an exponential $M(t) = 1000 e^{0.05 t}$ .
  ans
2. How much money will there be at year 30 (i.e. $t = 30$ )?
  $4481. 69
  ans
3. When will there be $2000$ dollars?
  $13.86$ years.
  ans
The mass of bacteria on a deceased animal follows the equation M′(t)=0.1M(t), where M(t) is measured in grams and t is measured in hours.
1. If $M(0) = 1$ , what is a function that satisfies this initial value problem?
  $M(t) = e^{0.1t}$
  ans
2. How much bacteria will there be at $t = 24$ ?
  $11.02$ grams
  ans
3. When will there be one kilogram of bacteria?
  2 days, 21 hours
  ans
For a cooling object outside in 0∘ degree weather, temperature decreases according to the differential equation T′(t)=−0.05T(t), where t is measured in minutes and T(t) measured in Fahrenheit.
1. If the temperature is initially $72^\circ$ , what is the function that satisfies this initial value problem?
  $T(t) = 72 e^{-0.05t}$
  ans
2. What is the temperature after 1/2 hour?
  $16.06$ degrees
  ans
3. At what time did the object reach the freezing point of water?
  Approximately $16$ minutes
  ans

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