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 \(M(0) = 1000\).
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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
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How much money will there be at year 30 (i.e. \(t = 30\))?
$4481. 69ans
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When will there be \(2000\) dollars?
\(13.86\) years.ans
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What is a function that satisfies this initial value problem?
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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.
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If \(M(0) = 1\), what is a function that satisfies this initial value problem?
\(M(t) = e^{0.1t}\)ans
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How much bacteria will there be at \(t = 24\)?
\(11.02\) gramsans
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When will there be one kilogram of bacteria?
2 days, 21 hoursans
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If \(M(0) = 1\), what is a function that satisfies this initial value problem?
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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.
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If the temperature is initially \(72^\circ\), what is the function that satisfies this initial value problem?
\(T(t) = 72 e^{-0.05t}\)ans
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What is the temperature after 1/2 hour?
\(16.06\) degreesans
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At what time did the object reach the freezing point of water?
Approximately \(16\) minutesans
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If the temperature is initially \(72^\circ\), what is the function that satisfies this initial value problem?