5.10: Homework- Growth and Decay

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Money that is compounded continuously follows the differential equation $M'(t) = r M(t)$, 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 $M'(t) = 0.1 M(t)$, 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 $0^\circ$ degree weather, temperature decreases according to the differential equation $T'(t) = -0.05 T(t)$, 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