A culture of bacteria has a population of 150150150 cells when it is first observed. The population doubles every 12 hr12~\text{hr}12 hr, which means its population is governed by the function p(t)=150⋅2t12p\left(t\right)=150\cdot{$$2^{\frac{t}$${12}}}p(t)=150⋅212t, where ttt is the number of hours after the first observation.How long does it take the population to triple in size?

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 1 - Functions

Problema 1.82d

Capitolo 1, Problema 1.82d

A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~$\text{hr}$$ , which means its population is governed by the function $p$\left$(t$\right$)=150$\cdot{2^{\frac{t}{12}$}}$ , where $t$ is the number of hours after the first observation.
How long does it take the population to triple in size?

Guida verificata passo dopo passo

Identify the initial population and the function governing the population growth. The initial population is 150 cells, and the function is given by $ p(t) = 150 \cdot 2^{\frac{t}{12}} $.

Determine the target population size for the bacteria to triple. Since the initial population is 150, the target population is $ 3 \times 150 = 450 $ cells.

Set up the equation to find the time $ t $ when the population reaches 450 cells: $ 450 = 150 \cdot 2^{\frac{t}{12}} $.

Divide both sides of the equation by 150 to isolate the exponential term: $ 3 = 2^{\frac{t}{12}} $.

Take the logarithm of both sides to solve for $ t $. Use the property of logarithms that $ \log(a^b) = b \cdot \log(a) $: $ \log(3) = \frac{t}{12} \cdot \log(2) $. Solve for $ t $ by multiplying both sides by 12 and dividing by $ \log(2) $.

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Exponential Growth

Exponential growth occurs when a quantity increases at a rate proportional to its current value. In the context of the bacteria population, the population doubles every 12 hours, which can be modeled by the function p(t) = 150 * 2^(t/12). This type of growth leads to rapid increases over time, making it essential to understand when analyzing population dynamics.

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Doubling Time

Doubling time is the period it takes for a quantity to double in size. For the bacteria population, the doubling time is given as 12 hours. This concept is crucial for calculating future population sizes and understanding the rate of growth, as it provides a consistent measure of how quickly the population expands.

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Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are used to solve for time in growth problems. To determine how long it takes for the population to triple, one can set up the equation 3 * 150 = 150 * 2^(t/12) and solve for t using logarithms. This concept is vital for transitioning from exponential growth equations to practical time calculations.

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A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~$\text{hr}$$ , which means its population is governed by the function $p$\left$(t$\right$)=150$\cdot{2^{\frac{t}{12}$}}$ , where $t$ is the number of hours after the first observation.

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