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

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

All textbooks

Briggs 3rd Edition

Ch. 1 - Functions

Problem 1.82c

Chapter 1, Problem 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?

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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.

Recommended video:

06:02

Work Done On A Spring (Hooke's Law) Example 1

Related Practice

Textbook Question

Identify the symmetry (if any) in the graphs of the following equations.

$y^2-4x^2=4$

views

Textbook Question

Inverse of composite functions

c. Explain why if g and h are one-to-one, the inverse of ƒ(x) = g(h(x)) exists.

Textbook Question

Inverse of composite functions

b. Let g(x) = x² + 1 and h(x) = √x. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹

Textbook Question

Composite functions

Let ƒ(x) = x³, g (x) = sin x and h(x) = √x .

Find the domain of ƒ o g.

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.

How long does it take the population to triple in size?

views

Textbook Question

Splitting up curves The unit circle x² + y² = 1 consists of four one-to-one functions, ƒ₁ (x), ƒ₂(x) , ƒ₃(x), and ƒ₄ (x) (see figure)<IMAGE>.

b. Find the inverse of each function and write it as y= ƒ⁻¹ (x)

views