# 5.10: Homework- Growth and Decay

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1. 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 $$M(t) = 1000 e^{0.05 t}$$.
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2. How much money will there be at year 30 (i.e. $$t = 30$$)?
\$4481. 69
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3. When will there be $$2000$$ dollars?
$$13.86$$ years.
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2. 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?
$$M(t) = e^{0.1t}$$
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2. How much bacteria will there be at $$t = 24$$?
$$11.02$$ grams
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3. When will there be one kilogram of bacteria?
2 days, 21 hours
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3. 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?
$$T(t) = 72 e^{-0.05t}$$
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2. What is the temperature after 1/2 hour?
$$16.06$$ degrees
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3. At what time did the object reach the freezing point of water?
Approximately $$16$$ minutes
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