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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.1.43a
Chapter 6, Problem 6.1.43a

40–43. Population growth


A culture of bacteria in a Petri dish has an initial population of 1500 cells and grows at a rate (in cells/day) of N′(t) = 100e^−0.25t. Assume t is measured in days.


a. What is the population after 20 days? After 40 days?

Verified step by step guidance
1
Identify the given rate of change of the population, which is the derivative of the population function: \(N\'(t) = 100e^{-0.25t}\), where \(t\) is in days.
Recognize that to find the population function \(N(t)\), you need to integrate the rate function \(N\'(t)\) with respect to \(t\): \(N(t) = \int 100e^{-0.25t} \, dt + C\).
Perform the integration of \(100e^{-0.25t}\): use the integral formula \(\int e^{kt} dt = \frac{1}{k} e^{kt} + C\), keeping track of the constant of integration \(C\).
Use the initial condition given, \(N(0) = 1500\), to solve for the constant \(C\) by substituting \(t=0\) into the integrated function and setting it equal to 1500.
Once you have the explicit formula for \(N(t)\), substitute \(t=20\) and \(t=40\) into this formula to find the population after 20 days and 40 days, respectively.

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Key Concepts

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

Derivative as Rate of Change

The derivative N′(t) represents the instantaneous rate of change of the population with respect to time. In this problem, N′(t) = 100e^−0.25t gives the growth rate in cells per day at any time t, which is essential for determining how the population changes over time.
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Intro To Related Rates

Definite Integral for Accumulated Change

To find the total change in population over a time interval, we integrate the rate function N′(t) from the start to the end time. The definite integral of N′(t) from 0 to t gives the net increase in population during that period.
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Definition of the Definite Integral

Initial Condition and Population Function

The initial population N(0) = 1500 serves as the starting value for the population function N(t). After integrating the rate function, adding this initial value gives the total population at any time t, allowing calculation of population after 20 and 40 days.
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Related Practice
Textbook Question

Emptying a conical tank A water tank is shaped like an inverted cone with height 6 m and base radius 1.5 m (see figure).

a. If the tank is full, how much work is required to pump the water to the level of the top of the tank and out of the tank?

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Textbook Question

Flow rates in the Spokane River The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functions

r1(t) = 0.25t²+37.46t+722.47 (April) and

r2(t) = 0.90t²−69.06t+2053.12 (June), where the discharge is measured in millions of cubic feet per day, and t=0 corresponds to the beginning of the first day of the month (see figure).

a. Determine the total amount of water that flows through Spokane in April (30 days). 

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Textbook Question

Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.


a. Verify that the amount of blood pumped over a one-second interval is 70 mL.

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Textbook Question

Region R is revolved about the line y=1 to form a solid of revolution.


a. What is the radius of a cross section of the solid at a point x in [0, 4]?

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Textbook Question

Consider the region R in the first quadrant bounded by y=x^1/n and y=x^n, where n>1 is a positive number.


a. Find the volume V(n) of the solid generated when R is revolved about the x-axis. Express your answer in terms of n.

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Textbook Question

For the given regions R₁ and R₂, complete the following steps.


a. Find the area of region R₁.


R₁ is the region in the first quadrant bounded by the y-axis and the curves y=2x^2 and y=3−x; R₂ is the region in the first quadrant bounded by the x-axis and the curves y=2x^2 and y=3−x(see figure).

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