Textbook Question
Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.
b. Find the displacement of the object on the interval 0≤t≤6.
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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 guidance1
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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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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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
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Winding a chain A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of 5kg/m.
a. How much work is required to wind the entire chain onto the cylinder using the winch?
Textbook Question
Find the area of the region (see figure) in two ways.
a. Using integration with respect to x.
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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.
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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Compressing and stretching a spring Suppose a force of 15 N is required to stretch and hold a spring 0.25 m from its equilibrium position.
b. How much work is required to compress the spring 0.2 m from its equilibrium position?