1.9.1 Use the idea of the weighted average to find the
following. 1.0 L of water at 30?C is mixed with 2.0 L of water at
100?C. What is the temperature of the resulting mixture?
Get solution
1.9.2 Use the idea of the weighted average to find the following. 2.0
ml of water with a salt concentration of 0.85 mol/L, is mixed with 5.0
ml of water with a salt concentration of 0.70 mol/L. What is the
concentration of the mixture?
Get solution
1.9.3
Use the idea of the weighted average to find the following. In a
class of 52 students, 20 scored 50 on a test, 18 scored 75, and the rest
scored 100. What was the average score?
Get solution
1.9.4 Use the idea of the weighted average to find the following. In
a class of 100 students, 10 score at 20, 20 score at 40, 30 score at
60, and 40 score at 80. What is the average score in the class?
Get solution
1.9.5 Express the following weighted averages in terms of the given variables. 1.0 L of water at temperature ... is mixed with 2.0 L of water at temperature .... What is the temperature of the resulting mixture? Set ... = 30 and ...
= 100 and compare with the result of Exercise 1. Exercise 1 1.0 L of
water at 30?C is mixed with 2.0 L of water at 100?C. What is the
temperature of the resulting mixture?
Get solution
1.9.6 Express the following weighted averages in terms of the given variables. ... liters of water at 30?C is mixed with ...liters of water at 100?C. What is the temperature of the resulting ixture? Set ... =1.0 and ...=2.0
and compare with the result of Exercise 1. Exercise 1 1.0 L of water
at 30?C is mixed with 2.0 L of water at 100?C. What is the temperature
of the resulting mixture?
Get solution
1.9.7 Express the following weighted averages in terms of the given variables. ... liters of water at temperature ...is mixed with ... liters of water at temperature .... What is the temperature of the resulting mixture?
Get solution
1.9.8 Express the following weighted averages in terms of the given variables. ... liters of water at temperature ... is mixed with ... liters of water at temperature ...liters of water at temperature .... What is the temperature of the resulting mixture?
Get solution
1.9.9
The following are similar to examples of weighted averages with
absorption. 1.0
L of water at 30?C is to be mixed with 2.0 L of water at 100?C, as in
Exercise 1. Before mixing, however, the temperature of each moves
half-way to 0 ?C (so the 30?Cwater cools to 15?C). What is the
temperature of the resulting mixture? Is this half the temperature of
the result in Exercise 1?Exercise 1 Use the idea of the weighted
average to find the following. 1.0 L of water at 30?C is mixed with 2.0
L of water at 100?C. What is the temperature of the resulting mixture?
Get solution
1.9.10
The following are similar to examples of weighted averages with
absorption. 2.0
ml of water with a salt concentration of 0.85 mol/L, is to be mixed
with 5.0 ml of water with a salt concentration of 0.70 mol/L, as in
Exercise 2. Before mixing, however, evaporation leads the each
concentration of each component to double. What is the concentration of
the mixture? Is it exactly twice the concentration found in Exercise
2?Exercise 2 Use the idea of the weighted average to find the
following. 2.0
ml of water with a salt concentration of 0.85 mol/L, is mixed with 5.0
ml of water with a salt concentration of 0.70 mol/L. What is the
concentration of the mixture?
Get solution
1.9.11
The following are similar to examples of weighted averages with
absorption. In
a class of 52 students, 20 scored 50 on a test, 18 scored 75, and the
rest scored 100. The professor suspects cheating, however, and deducts
10 from each score. What is the average score after the deduction? Is it
exactly 10 less than the average found in Exercise 3? Exercise 3Use
the idea of the weighted average to find the following. In a class of
52 students, 20 scored 50 on a test, 18 scored 75, and the rest scored
100. What was the average score?
Get solution
1.9.12
The following are similar to examples of weighted averages with
absorption. In
a class of 100 students, 10 score at 20, 20 score at 40, 30 score at
60, and 40 score at 80 as in Exercise 4. Because students did so poorly,
the professor moves each score half way up toward 100 (so the students
with 20 are moved up to 60). What is the average score in the class? Is
the new average the old average moved half way to 100? Exercise 4Use
the idea of the weighted average to find the following. In
a class of 100 students, 10 score at 20, 20 score at 40, 30 score at
60, and 40 score at 80. What is the average score in the class?
Get solution
1.9.13 Suppose that the volume of the lungs is V , the amount breathed in and out is W, and the ambient concentration is γ mmol/L. For each of the given sets of parameter values and the given initial condition, find the following:
a. The amount of chemical in the lungs before breathing
b. The amount of chemical breathed out
c. The amount of chemical in the lungs after breathing out
d. The amount of chemical breathed in
e. The amount of chemical in the lungs after breathing in
f. The concentration of chemical in the lungs after breathing in g.
Compare this result with the result of using the general lungs
discrete-time dynamical system (equation 1.9.1). Remember that q = W/ V V = 2.0 L, W = 0.5 L, γ = 5.0 mmol/L, ... = 1.0 mmol/L.
Get solution
1.9.14 Suppose that the volume of the lungs is V , the amount breathed in and out is W, and the ambient concentration is γ mmol/L. For each of the given sets of parameter values and the given initial condition, find the following:
a. The amount of chemical in the lungs before breathing
b. The amount of chemical breathed out
c. The amount of chemical in the lungs after breathing out
d. The amount of chemical breathed in
e. The amount of chemical in the lungs after breathing in
f. The concentration of chemical in the lungs after breathing in g.
Compare this result with the result of using the general lungs
discrete-time dynamical system (equation 1.9.1). Remember that q = W/ V V =1.0 L, W = 0.1 L, γ = 8.0 mmol/L, ... = 4.0 mmol/L.
Get solution
1.9.15 Suppose that the volume of the lungs is V , the amount breathed in and out is W, and the ambient concentration is γ mmol/L. For each of the given sets of parameter values and the given initial condition, find the following:
a. The amount of chemical in the lungs before breathing
b. The amount of chemical breathed out
c. The amount of chemical in the lungs after breathing out
d. The amount of chemical breathed in
e. The amount of chemical in the lungs after breathing in
f. The concentration of chemical in the lungs after breathing in g.
Compare this result with the result of using the general lungs
discrete-time dynamical system (equation 1.9.1). Remember that q = W/ V V =1.0 L, W = 0.9 L, γ = 5.0 mmol/L, ... = 9.0 mmol/L.
Get solution
1.9.16 Suppose that the volume of the lungs is V , the amount breathed in and out is W, and the ambient concentration is γ mmol/L. For each of the given sets of parameter values and the given initial condition, find the following:
a. The amount of chemical in the lungs before breathing
b. The amount of chemical breathed out
c. The amount of chemical in the lungs after breathing out
d. The amount of chemical breathed in
e. The amount of chemical in the lungs after breathing in
f. The concentration of chemical in the lungs after breathing in g.
Compare this result with the result of using the general lungs
discrete-time dynamical system (equation 1.9.1). Remember that q = W/ V V =10.0 L, W = 0.2 L, γ = 1.0 mmol/L, ... = 9.0 mmol/L.
Get solution
1.9.17 Find
and graph the updating function in the following cases. Cobweb for
three steps starting from the points indicated in the earlier problems.
Sketch the solutions. The situation in Exercise 13. Reference Exercise 13. V = 2.0 L, W = 0.5 L, γ = 5.0 mmol/L, ... = 1.0 mmol/L.
Get solution
1.9.18 Find
and graph the updating function in the following cases. Cobweb for
three steps starting from the points indicated in the earlier problems.
Sketch the solutions. The situation in Exercise 14. Reference Exercise 14. V =1.0 L, W = 0.1 L, γ = 8.0 mmol/L, ... = 4.0 mmol/L.
Get solution
1.9.19 Find
and graph the updating function in the following cases. Cobweb for
three steps starting from the points indicated in the earlier problems.
Sketch the solutions. The situation in Exercise 15. Reference Exercise 15 V =1.0 L, W = 0.9 L, γ = 5.0 mmol/L, ... = 9.0 mmol/L.
Get solution
1.9.20 Find
and graph the updating function in the following cases. Cobweb for
three steps starting from the points indicated in the earlier problems.
Sketch the solutions. The situation in Exercise 16. Reference Exercise 16 V =10.0 L, W = 0.2 L, γ = 1.0 mmol/L, ... = 9.0 mmol/L.
Get solution
1.9.21 Compute the equilibrium of the lungs discrete-time dynamical system and check that c∗ = γ . V = 2.0 L, W = 0.5 L, γ = 5.0 mmol/L, ... = 1.0 mmol/L (as in Exercise 13). Exercise 13 V = 2.0 L, W = 0.5 L, γ = 5.0 mmol/L, ... = 1.0 mmol/L.
Get solution
1.9.22 Compute the equilibrium of the lungs discrete-time dynamical system and check that c∗ = γ . V =1.0 L, W = 0.1 L, γ = 8.0 mmol/L, ... = 4.0 mmol/L (as in Exercise 14). Reference Exercise 14. V =1.0 L, W = 0.1 L, γ = 8.0 mmol/L, ... = 4.0 mmol/L.
Get solution
1.9.23 Compute the equilibrium of the lungs discrete-time dynamical system and check that c∗ = γ . V =1.0 L, W = 0.9 L, γ = 5.0 mmol/L, ...= 9.0 mmol/L (as in Exercise 15). Reference Exercise 15 V =1.0 L, W = 0.9 L, γ = 5.0 mmol/L, ... = 9.0 mmol/L.
Get solution
1.9.24 Compute the equilibrium of the lungs discrete-time dynamical system and check that c∗ = γ . V =10.0 L, W = 0.2 L, γ = 1.0 mmol/L, ... = 9.0 mmol/L (as in Exercise 16). Reference Exercise 16 V =10.0 L, W = 0.2 L, γ = 1.0 mmol/L, ... = 9.0 mmol/L.
Get solution
1.9.25 Compare the equilibrium and total amount absorbed per breath for different values of q. Use an ambient concentration of γ = 0.21 and a volume of V = 6.0L. Suppose q = 0.4 and α = 0.1. Why is the equilibrium concentration higher than with q = 0.2 even though the person is breathing more?
Get solution
1.9.26 Compare the equilibrium and total amount absorbed per breath for different values of q. Use an ambient concentration of γ = 0.21 and a volume of V = 6.0L. Suppose q = 0.1 and α = 0.05.
Think of this as a person gasping for breath. Why is the concentration
nearly the same as in Example 1.9.9? Does this mean that gasping for
breath is OK?
Get solution
1.9.27 The following problems investigate absorption that is not proportional to the concentration in the lungs. Assume γ = 0.21 and q = 0.1, and find the equilibrium concentration. Oxygen
concentration is reduced by 2% each breath (that is, if the
concentration before absorption were 18%, it would be 16% after
absorption). Find the discrete-time dynamical system and the
equilibrium. Are there values of ... for which the system does not make sense?
Get solution
1.9.28 The following problems investigate absorption that is not proportional to the concentration in the lungs. Assume γ = 0.21 and q = 0.1, and find the equilibrium concentration. Oxygen
concentration is reduced by 1% each breath. Find the discrete-time
dynamical system and the equilibrium. Are there values of ...for which the system does not make sense?
Get solution
1.9.29 The following problems investigate absorption that is not proportional to the concentration in the lungs. Assume γ = 0.21 and q = 0.1,
and find the equilibrium concentration. The amount absorbed is ...
This models a case where the only oxygen available is that in excess of
the concentration in the blood, which is roughly 5%.
Get solution
1.9.30 The following problems investigate absorption that is not proportional to the concentration in the lungs. Assume γ = 0.21 and q = 0.1, and find the equilibrium concentration. The amount absorbed is ...
Get solution
1.9.31 Find the value of the parameter that produces an exhaled concentration of exactly 0.15. Assume γ = 0.21 and q = 0.1. Oxygen concentration is reduced by an amount A (generalizing the case in Exercises 27 and 28). How does the amount absorbed with this value of A compare with the amount of oxygen absorbed in Example 1.9.9? Reference Example 1.9.9 ...
Get solution
1.9.32 Find the value of the parameter that produces an exhaled concentration of exactly 0.15. Assume γ = 0.21 and q = 0.1. The amount absorbed is α(... − 0.05) (generalizing
the case where only available oxygen is absorbed in Exercises 29 and
30). How does the amount absorbed with this value of α compare with the amount of oxygen absorbed in Example 1.9.9? Reference Example 1.9.9 ...
Get solution
1.9.33 The
following problems investigate production of carbon dioxide by the
lungs. Suppose that the concentration increases by an amount S before the air is exchanged. Assume an ambient concentration of carbon dioxide of γ = 0.0004 and q = 0.1. Suppose S = 0.001. Write the discrete-time dynamical system and find its equilibrium. Compare the equilibrium with the ambient concentration.
Get solution
1.9.34 The
following problems investigate production of carbon dioxide by the
lungs. Suppose that the concentration increases by an amount S before the air is exchanged. Assume an ambient concentration of carbon dioxide of γ = 0.0004 and q = 0.1. The
actual concentration of carbon dioxide in exhaled air is about 0.04, or
100 times the ambient concentration. Find the value of S that gives this as the equilibrium.
Get solution
1.9.35 A bacterial population that has per capita production r < 1
but that is supplemented each generation follows a discrete-time
dynamical system much like that of the lungs. Use the following steps to
build the discrete-time dynamical system.
a. Starting from 3.0×... bacteria, find the number after reproduction.
b. Find the number after the new bacteria are added.
c. Find the discrete-time dynamical system. A population of bacteria has per capita production r = 0.6, and 1.0 ×... bacteria are added each generation.
Get solution
1.9.36 A bacterial population that has per capita production r < 1
but that is supplemented each generation follows a discrete-time
dynamical system much like that of the lungs. Use the following steps to
build the discrete-time dynamical system.
a. Starting from 3.0×... bacteria, find the number after reproduction.
b. Find the number after the new bacteria are added.
c. Find the discrete-time dynamical system. A population of bacteria has per capita production r = 0.2, and 5.0 ×...bacteria are added each generation.
Get solution
1.9.37
Find the equilibrium population of bacteria in the following cases
with supplementation. A population of bacteria has per capita
production r = 0.6, and 1.0×... bacteria are added each generation (as in Exercise 35). Exercise 35 A population of bacteria has per capita production r = 0.6, and 1.0 ×... bacteria are added each generation.
Get solution
1.9.38
Find the equilibrium population of bacteria in the following cases
with supplementation. A population of bacteria has per capita
production r = 0.2, and 5.0×... bacteria are added each generation (as in Exercise 36). Exercise 36 A population of bacteria has per capita production r = 0.2, and 5.0 ×...bacteria are added each generation.
Get solution
1.9.39
Find the equilibrium population of bacteria in the following cases
with supplementation. A population of bacteria has per capita
production r = 0.5, and S bacteria are added each generation. What happens to the equilibrium when S is large? Does this make biological sense?
Get solution
1.9.40
Find the equilibrium population of bacteria in the following cases
with supplementation. A population of bacteria has per capita
production r < 1, and 1.0 ×... bacteria are added each generation. What happens to the equilibrium if r = 0? What happens if r is close to 1? Do these results make biological sense?
Get solution
1.9.41 Lakes
receive water from streams each year and lose water to outflowing
streams and evaporation. The following values are based on the Great
Salt Lake in Utah. The lake receives 3.0 ×... of water per year with salinity of 1 part per thousand (concentration 0.001). The lake contains 3.3 × 1 ...
of water and starts with no salinity. Assume that the water that flows
out has a concentration equal to that of the entire lake. Compute the
discrete-time dynamical system by finding (a) the total salt before the
inflow, (b) total water, (c) total salt and salt concentration after
inflow, and (d) total water, total salt, and salt concentration after
outflow or evaporation, There is no evaporation, and 3.0 ×... of water flows out each year.
Get solution
1.9.42 Lakes
receive water from streams each year and lose water to outflowing
streams and evaporation. The following values are based on the Great
Salt Lake in Utah. The lake receives 3.0 ×... of water per year with salinity of 1 part per thousand (concentration 0.001). The lake contains 3.3 × 1 ...
of water and starts with no salinity. Assume that the water that flows
out has a concentration equal to that of the entire lake. Compute the
discrete-time dynamical system by finding (a) the total salt before the
inflow, (b) total water, (c) total salt and salt concentration after
inflow, and (d) total water, total salt, and salt concentration after
outflow or evaporation, 1.5×... of water flows out each year, and 1.5×... evaporates. No salt is lost through evaporation.
Get solution
1.9.43 Lakes
receive water from streams each year and lose water to outflowing
streams and evaporation. The following values are based on the Great
Salt Lake in Utah. The lake receives 3.0 ×... of water per year with salinity of 1 part per thousand (concentration 0.001). The lake contains 3.3 × 1 ...
of water and starts with no salinity. Assume that the water that flows
out has a concentration equal to that of the entire lake. Compute the
discrete-time dynamical system by finding (a) the total salt before the
inflow, (b) total water, (c) total salt and salt concentration after
inflow, and (d) total water, total salt, and salt concentration after
outflow or evaporation, A total of 3.0×... of water evaporates, and there is no outflow.
Get solution
1.9.44 Lakes
receive water from streams each year and lose water to outflowing
streams and evaporation. The following values are based on the Great
Salt Lake in Utah. The lake receives 3.0 ×... of water per year with salinity of 1 part per thousand (concentration 0.001). The lake contains 3.3 × 1 ...
of water and starts with no salinity. Assume that the water that flows
out has a concentration equal to that of the entire lake. Compute the
discrete-time dynamical system by finding (a) the total salt before the
inflow, (b) total water, (c) total salt and salt concentration after
inflow, and (d) total water, total salt, and salt concentration after
outflow or evaporation, Assume instead that 2.0×... of water evaporates and there is no outflow. The volume of this lake is increasing.
Get solution
1.9.45
Find
the equilibrium concentration of salt in a lake in the following cases.
Describe the result in words by comparing the equilibrium salt level
with the salt level of the water flowing in. The situation described in
Exercise 41. Reference Exercise 41 There is no evaporation, and 3.0 ×... of water flows out each year.
Get solution
1.9.46 Find
the equilibrium concentration of salt in a lake in the following cases.
Describe the result in words by comparing the equilibrium salt level
with the salt level of the water flowing in. The situation described in Exercise 42. Reference Exercise 42 1.5×... of water flows out each year, and 1.5×... evaporates. No salt is lost through evaporation.
Get solution
1.9.47 Find
the equilibrium concentration of salt in a lake in the following cases.
Describe the result in words by comparing the equilibrium salt level
with the salt level of the water flowing in. The situation described in Exercise 43. Reference Exercise 43 A total of 3.0×... of water evaporates, and there is no outflow.
Get solution
1.9.48
lab is growing and harvesting a culture of valuable bacteria
described by the discrete-time dynamical system ... The bacteria
have per capita production r , and h bacteria are harvested each generation.
Get solution
1.9.49 Suppose that r = 1.5 and h = 1.0 ×... bacteria. Sketch the updating function, and find the equilibrium both algebraically and graphically.
Get solution
1.9.50 Without setting r and h to particular values, find the equilibrium algebraically. Does the equilibrium get larger when h gets larger? Does it get larger when r gets larger? If the answers seem odd (as they should), look at a cobweb diagram to try to figure out why.
Get solution
1.9.51 Investigate which factor is most important in absorbing oxygen at the maximum rate: the volume V of the lungs, the amount exchanged W, or the fraction absorbed α,
using Equation 1.9.2. If an athlete could train to increase one of
these values, which would be the most effective? Equation 1.9.2 Total
absorbed per breath = αc∗V.
Get solution
