- Money that is compounded continuously follows the differential equation [latex]M'(t) = r M(t)[/latex], where [latex]t[/latex] is measured in years, [latex]M(t)[/latex] is measured in dollars, and [latex]r[/latex] is the rate. Suppose [latex]r = 0.05[/latex] and [latex]M(0) = 1000[/latex].
- What is a function that satisfies this initial value problem?
We know from class that this is an exponential [latex]M(t) = 1000 e^{0.05 t}[/latex].ans
- How much money will there be at year 30 (i.e. [latex]t = 30[/latex])?
$4481. 69ans
- When will there be [latex]2000[/latex] dollars?
[latex]13.86[/latex] years.ans
- What is a function that satisfies this initial value problem?
- The mass of bacteria on a deceased animal follows the equation [latex]M'(t) = 0.1 M(t)[/latex], where [latex]M(t)[/latex] is measured in grams and [latex]t[/latex] is measured in hours.
- If [latex]M(0) = 1[/latex], what is a function that satisfies this initial value problem?
[latex]M(t) = e^{0.1t}[/latex]ans
- How much bacteria will there be at [latex]t = 24[/latex]?
[latex]11.02[/latex] gramsans
- When will there be one kilogram of bacteria?
2 days, 21 hoursans
- If [latex]M(0) = 1[/latex], what is a function that satisfies this initial value problem?
- For a cooling object outside in [latex]0^\circ[/latex] degree weather, temperature decreases according to the differential equation [latex]T'(t) = -0.05 T(t)[/latex], where [latex]t[/latex] is measured in minutes and [latex]T(t)[/latex] measured in Fahrenheit.
- If the temperature is initially [latex]72^\circ[/latex], what is the function that satisfies this initial value problem?
[latex]T(t) = 72 e^{-0.05t}[/latex]ans
- What is the temperature after 1/2 hour?
[latex]16.06[/latex] degreesans
- At what time did the object reach the freezing point of water?
Approximately [latex]16[/latex] minutesans
- If the temperature is initially [latex]72^\circ[/latex], what is the function that satisfies this initial value problem?