Consider a population of bacteria that grows according to the function [latex]f(t)=500{e}^{0.05t},[/latex] where [latex]t[/latex] is measured in minutes. How many bacteria are present in the population after [latex]4[/latex] hours? When does the population reach [latex]100[/latex] million bacteria?
There are [latex]81,377,396[/latex] bacteria in the population after [latex]4[/latex] hours. The population reaches [latex]100[/latex] million bacteria after [latex]244.12[/latex] minutes.
Suppose instead of investing at age [latex]25\sqrt{{b}^{2}-4ac},[/latex] the student waits until age 35. How much would she have to invest at [latex]5\text{%}?[/latex] At [latex]6\text{%}?[/latex]
At [latex]5\%[/latex] interest, she must invest [latex]$223,130.16.[/latex] At [latex]6\%[/latex] interest, she must invest [latex]$165,298.89.[/latex]
Suppose it takes [latex]9[/latex] months for the fish population in the last example to reach [latex]1000[/latex] fish. Under these circumstances, how long do the owner’s friends have to wait?
[latex]38.90[/latex] months
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A bacteria culture starts with [latex]1000[/latex] cells and grows exponentially. After [latex]2[/latex] hours, there are [latex]1500[/latex] cells. How many cells will be present after [latex]5[/latex] hours?
Use the exponential growth model: [latex]y = y_0 e^{kt}[/latex]
We know:
[latex]y_0 = 1000[/latex] (initial number of cells)
After [latex]2[/latex] hours, [latex]y = 1500[/latex]
Find [latex]k[/latex] using the [latex]2[/latex]-hour data point:
Suppose the room is warmer [latex](75\text{°}\text{F})[/latex] and, after [latex]2[/latex] minutes, the coffee has cooled only to [latex]185\text{°}\text{F}.[/latex] When is the coffee first cool enough to serve? When is the coffee be too cold to serve? Round answers to the nearest half minute.
The coffee is first cool enough to serve about [latex]3.5[/latex] minutes after it is poured. The coffee is too cold to serve about [latex]7[/latex] minutes after it is poured.
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If we have [latex]100[/latex] g of [latex]\text{carbon-}14,[/latex] how much is left after [latex]t[/latex] years? If an artifact that originally contained [latex]100[/latex] g of carbon now contains [latex]20g[/latex] of carbon, how old is it? Round the answer to the nearest hundred years.
A total of [latex]94.13[/latex] g of carbon remains. The artifact is approximately [latex]13,300[/latex] years old.
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A radioactive sample initially contains [latex]100[/latex] grams of a certain isotope. After [latex]30[/latex] days, the sample contains [latex]80[/latex] grams of the isotope. What is the half-life of this isotope?
Use the exponential decay model: [latex]y = y_0 e^{-kt}[/latex]
We know:
[latex]y_0 = 100[/latex] (initial amount)
After [latex]30[/latex] days, [latex]y = 80[/latex]