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Chapter 7: Problem 9

A bacterial culture starts with 2000 bacteria and doubles every \(0.75 \mathrm{h}\). After how many hours will the bacteria count be 32 000?

Short Answer

Expert verified
It takes 3 hours for the bacteria count to reach 32,000.

Step by step solution

01

Understand the problem

The problem involves exponential growth where the bacterial culture starts with 2000 bacteria and doubles every 0.75 hours. We need to find the time it takes for the bacteria count to reach 32,000.
02

Define the exponential growth formula

The formula for exponential growth is given by \[ N(t) = N_0 \times 2^{(t/T)} \]where:- \(N(t)\) is the number of bacteria at time \(t\)- \(N_0\) is the initial number of bacteria- \(T\) is the doubling time- \(t\) is the time in hours.
03

Substitute the known values into the formula

Given:\( N_0 = 2000 \)\( T = 0.75 \) hours \( N(t) = 32000 \)Substitute these values into the formula:\[ 32000 = 2000 \times 2^{(t/0.75)} \]
04

Solve for \(t\)

First, divide both sides by 2000 to isolate the exponential term:\[ \frac{32000}{2000} = 2^{(t/0.75)} \]\[ 16 = 2^{(t/0.75)} \]Next, express 16 as a power of 2:\[ 16 = 2^4 \]So we have:\[ 2^4 = 2^{(t/0.75)} \]Since the bases are the same, we can set the exponents equal to each other:\[ 4 = \frac{t}{0.75} \]Multiply both sides by 0.75:\[ t = 4 \times 0.75 \]\[ t = 3 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Doubling Time
In the context of bacterial growth, doubling time is the period it takes for a population of bacteria to double in number. Knowing the doubling time helps predict how quickly a bacterial population can grow under ideal conditions. For example, if we start with 2000 bacteria and the doubling time is 0.75 hours, we know that in 0.75 hours, the bacteria count will be 4000, in another 0.75 hours it will be 8000, and so on. This predictable pattern allows us to calculate how long it will take to reach any given population size. Doubling time is a critical concept in understanding exponential growth.
Exponential Growth Formula
Exponential growth describes how a quantity increases rapidly in proportion to its current value. For bacterial growth, the exponential growth formula is: \[ N(t) = N_0 \times 2^{(t/T)} \] Where:
  • \(N(t)\): the number of bacteria at time \(t\)
  • \(N_0\): the initial number of bacteria
  • \(T\): the doubling time
  • \(t\): the time in hours
This formula helps us to understand and predict the growth of a bacterial population over time. By plugging in different values, we can determine how long it will take for the bacteria to reach a certain number.
Isolate Exponential Term
To solve exponential growth problems, it's often necessary to isolate the exponential term. In the given problem, the equation becomes:\[ 32000 = 2000 \times 2^{(t/0.75)} \]We start by dividing both sides by 2000:\[ \frac{32000}{2000} = 2^{(t/0.75)} \]This simplifies to:\[ 16 = 2^{(t/0.75)} \]Next, express 16 as a power of 2:\[ 16 = 2^4 \]Now, equate the exponents because the bases are the same:\[ 4 = \frac{t}{0.75} \]Finally, solve for \(t\) by multiplying both sides by 0.75:\[ t = 4 \times 0.75 \]This gives us \(t = 3\) hours. So, it takes 3 hours for the bacterial population to grow from 2000 to 32,000. Isolating the exponential term is crucial in solving these equations effectively.

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Most popular questions from this chapter

If seafood is not kept frozen (below \(0^{\circ} \mathrm{C}\) ), it will spoil due to bacterial growth. The relative rate of spoilage increases with temperature according to the model \(R=100(2.7)^{\frac{T}{s}},\) where \(T\) is the temperature, in degrees Celsius, and \(R\) is the relative spoilage rate. a) Sketch a graph of the relative spoilage rate \(R\) versus the temperature \(T\) from \(0^{\circ} \mathrm{C}\) to \(25^{\circ} \mathrm{C}\) b) Use your graph to predict the temperature at which the relative spoilage rate doubles to \(200 .\) c) What is the relative spoilage rate at \(15^{\circ} \mathrm{C} ?\) d) If the maximum acceptable relative spoilage rate is \(500,\) what is the maximum storage temperature?

Money in a savings account earns compound interest at a rate of \(1.75 \%\) per year. The amount, \(A,\) of money in an account can be modelled by the exponential function \(A=P(1.0175)^{n}\) where \(P\) is the amount of money first deposited into the savings account and \(n\) is the number of years the money remains in the account. a) Graph this function using a value of \(P=\$ 1\) as the initial deposit. b) Approximately how long will it take for the deposit to triple in value? c) Does the amount of time it takes for a deposit to triple depend on the value of the initial deposit? Explain. d) In finance, the rule of 72 is a method of estimating an investment's doubling time when interest is compounded annually. The number 72 is divided by the annual interest rate to obtain the approximate number of years required for doubling. Use your graph and the rule of 72 to approximate the doubling time for this investment.

Statistics indicate that the world population since 1995 has been growing at a rate of about \(1.27 \%\) per year. United Nations records estimate that the world population in 2011 was approximately 7 billion. Assuming the same exponential growth rate, when will the population of the world be 9 billion?

a) On the same set of axes, sketch the graph of the function \(y=5^{x},\) and then sketch the graph of the inverse of the function by reflecting its graph in the line \(y=x\) b) How do the characteristics of the graph of the inverse of the function relate to the characteristics of the graph of the original exponential function? c) Express the equation of the inverse of the exponential function in terms of \(y\) That is, write \(x=F(y)\)

Simionie needs S7000 to buy a snowmobile, but only has \(\$ 6000 .\) His bank offers a GIC that pays an annual interest rate of \(3.93 \%,\) compounded annually. How long would Simionie have to invest his money in the GIC to have enough money to buy the snowmobile?
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