Chapter 3: Q10E (page 102)
Solve Problem 9 ifgrams of chemical
is present initially. At what time is the chemical
half-formed?
Short Answer
Answer
The limiting chemicalis
The time at which the chemical C is half-formed
Chapter 3: Q10E (page 102)
Solve Problem 9 ifgrams of chemical
is present initially. At what time is the chemical
half-formed?
Answer
The limiting chemicalis
The time at which the chemical C is half-formed
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Get started for freeWe have not discussed methods by which systems of first-order differential equations can be solved. Nevertheless, systems such as (2) can be solved with no knowledge other than how to solve a single linear first-order equation. Find a solution of ( 2 ) subject to the initial conditions.
Use Euler’s method with step size h = 0.1 to approximate y (1.2), where y(x) is a solution of the initial-value problem .
Newton’s Law of Cooling/Warming As shown in Figure 3.3.12, a small metal bar is placed inside container A, and container A then is placed within a much larger container B. As the metal bar cools, the ambient temperature of the medium within container A changes according to Newton’s law of cooling. As container A cools, the temperature of the medium inside container B does not change significantly and can be considered to be a constant TB. Construct a mathematical model for the temperatures and , where is the temperature of the metal bar inside container A. As in Problems 1, 5, and 20, this model can be solved by using prior knowledge. Find a solution of the system subject to the initial conditions
The radioactive isotope of lead, , decays at a rateproportional to the amount present at time tand has a half-life of 3.3hours. If 1gram of this isotope is present initially, howlong will it take for 90% of the lead to decay?
How High?—No Air Resistance Suppose a small cannonball weighing 16 pounds is shot vertically upward, as shown in Figure 3.1.13, with an initial velocity v0 5 300 ft/s. The answer to the question “How high does the cannonball go?” depends on whether we take air resistance into account.
(a) Suppose air resistance is ignored. If the positive direction is upward, then a model for the state of the cannonball is given by d2 sydt2 5 2g (equation (12) of Section 1.3). Since dsydt 5 v(t) the last differential equation is the same as dvydt 5 2g, where we take g 5 32 ft/s2 . Find the velocity v(t) of the cannonball at time t.
(b) Use the result obtained in part (a) to determine the height s(t) of the cannonball measured from ground level. Find the maximum height attained by the cannonball.
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