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

Welcome back everyone. A certain species of fish in a lake reproduces at a rate that doubles its population every 24 hours. The initial population of the fish was 200. If the population T hours after the initial observation is represented by the function P of T equals 200 multiplied by T to the power of T divided by 24. What would be the population? Three days after the initial observation, there are four answer choices. A 800 B 1600 C 3000, 200 D 400. So essentially we're given our function P of T. Let's define it, it is equal to 200 multiplied by T to the power of T divided by 24 where T is given in hours in the problem. It also states that tea time is equal to three days because time is defined in hours, we have to convert it into hours. So first of all, let's use a conversion factor. We know that one day is equal to 24 hours. And therefore, when we perform the calculation, while we notice that the units cancel each other out, we cancel out days and we end up with hours. So three days will be three multiplied by 24 that's 72 hours. So we have the time value that we can use in the given function. And we want to evaluate P of 72 the population value after 72 hours. So we substitute 200 multiplied by T to the power of 72 divided by 24. And we simply have to perform the calculation. So that would be 200 multiplied by T to the power of 72 divided by 24 gives us three, two, cubed is equal to eight. So we end up with 200 multiplied by eight, which is equal to 1600. So that's our final answer. And we can state that the final answer to this problem is B 1600. Thank you for watching.

0. Functions

Exponential Functions

Calculus

0. Functions

Exponential Functions

2:38 minutes

Problem 1.82c

Textbook Question

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?

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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