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.What is the population 4 days4~\text{days}4 days after the first observation?

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 1 - Functions

문제 1.82c

1장, 문제 1.82c

A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~$\text{hr}$$ , which means its population is governed by the function $p$\left$(t$\right$)=150$\cdot{2^{\frac{t}{12}$}}$ , where $t$ is the number of hours after the first observation.
What is the population $4~$\text{days}$$ after the first observation?

검증된 단계별 안내

First, convert the time from days to hours since the function p(t) uses hours as the time unit. There are 24 hours in a day, so 4 days is equivalent to 4 * 24 = 96 hours.

Next, substitute t = 96 into the population function p(t) = 150 $\cdot$ 2^{$\frac{t}{12}$} to find the population after 96 hours.

The expression becomes p(96) = 150 $\cdot$ 2^{$\frac{96}{12}$}.

Simplify the exponent $\frac{96}{12}$ to get 8, so the expression becomes p(96) = 150 $\cdot$ 2^8.

Finally, calculate 2^8 and multiply the result by 150 to find the population after 4 days.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Growth

Exponential growth occurs when the increase in a quantity is proportional to its current value, leading to rapid growth over time. In this context, the bacteria population doubles every 12 hours, which can be modeled by an exponential function. This type of growth is characterized by a constant doubling time, making it crucial for understanding how populations expand in biological systems.

Population Function

The population function, represented as p(t) = 150·2^(t/12), describes the number of bacteria at any given time t in hours. The initial population is 150, and the function incorporates the doubling behavior of the population every 12 hours. Understanding this function is essential for calculating the population at specific time intervals, such as 4 days after the initial observation.

Time Conversion

Time conversion is necessary when dealing with different units of time, such as hours and days. In this problem, 4 days must be converted into hours to use the population function correctly. Since there are 24 hours in a day, 4 days equals 96 hours, which allows for accurate calculations of the bacteria population at that time.