EBK DIFFERENTIAL EQUATIONS
5th Edition
ISBN: 9780321974235
Author: Calvis
Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)
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Chapter 5.2, Problem 35P
Program Plan Intro
Program Description:Purpose of problem is to find the amount of salt in three tank at time t , the limiting amount of salt in each water tank and also draw a graph of
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Consider the two tanks shown in the figure below. Assume that tank A contains 50 gallons of water in which 25 pounds of salt is dissolved. Suppose tank B contains 50 gallons of pure water. Liquid is pumped into
and out of the tanks as indicated in the figure; the mixture exchanged between the two tanks and the liquid pumped out of tank B are assumed to be well stirred. We wish to construct a mathematical model that
describes the number of pounds x₁(t) and x₂(t) of salt in tanks A and B, respectively, at time t.
dx₁
dt
dx₂
dt
mixture
4 gal/min
This system is described by the system of equations
1
50 2
2
25
2
dx₁
dt
dx2
dt
=
=
pure water
3 gal/min
=
2
25
2
110
25
A
+
1
1
mixture
1 gal/min
B
with initial conditions x₁(0) = 25, x₂(0) = 0 (see (3) and the surrounding discussion on mixtures on page 107). What is the system of differential equations if, instead of pure water, a brine solution containing 4
pounds of salt per gallon is pumped into tank A?
mixture
3 gal/min
Consider the two tanks shown in the figure below. Assume that tank A contains 50 gallons of water in which 25 pounds of
salt is dissolved. Suppose tank B contains 50 gallons of pure water. Liquid is pumped into and out of the tanks as indicated
in the figure; the mixture exchanged between the two tanks and the liquid pumped out of tank B are assumed to be well
stirred. We wish to construct a mathematical model that describes the number of pounds x₁(t) and x₂(t) of salt in tanks A
and B, respectively, at time t.
dx₁1
dt
dx1
dt
pure water
3 gal/min
mixture
4 gal/min
This system is described by the system of equations
1
50
2
25
dx2
dt
2
25
2
25*1
1
+
-X2
mixture
1 gal/min
-X2
B
with initial conditions x₁(0) = 25, x₂(0) = 0 (see (3) and the surrounding discussion on mixtures on page 107). What is the
system of differential equations if, instead of pure water, a brine solution containing 3 pounds of salt per gallon is pumped
into tank A?
-2
25*1 + 50x2+6×
dx2
2 - (25)*₁- (25)×₂2 -
=
x2
dt…
Question 1
John currently has a shipment with the below details:
In this shipment there are two types of products together
Product 1
Dimension 371 x 25w x 27h
Total quantity 330
Gross weight 16kg
Average price: 40USD
Product 2
Dimension 541 x 38w x 25h
Total quantity 90
Gross weight 21.5kg
Average price 130USD
Calculate the following
i. What would be gross weight
ii.
Total volume
iii.
What would be combined price USD
iv.
Average number of pallets if pallet is 100*120 cm and height doesn't exceed 200cm
Chapter 5 Solutions
EBK DIFFERENTIAL EQUATIONS
Ch. 5.1 - Let A=[2347] and B=[3451]. Find (a) 2A+3B; (b)...Ch. 5.1 - Prob. 2PCh. 5.1 - Find AB and BA given A=[203415] and B=[137032].Ch. 5.1 - Prob. 4PCh. 5.1 - Prob. 5PCh. 5.1 - Prob. 6PCh. 5.1 - Prob. 7PCh. 5.1 - Prob. 8PCh. 5.1 - Prob. 9PCh. 5.1 - Prob. 10P
Ch. 5.1 - Prob. 11PCh. 5.1 - Prob. 12PCh. 5.1 - Prob. 13PCh. 5.1 - Prob. 14PCh. 5.1 - Prob. 15PCh. 5.1 - Prob. 16PCh. 5.1 - Prob. 17PCh. 5.1 - Prob. 18PCh. 5.1 - Prob. 19PCh. 5.1 - Prob. 20PCh. 5.1 - Prob. 21PCh. 5.1 - Prob. 22PCh. 5.1 - Prob. 23PCh. 5.1 - Prob. 24PCh. 5.1 - Prob. 25PCh. 5.1 - Prob. 26PCh. 5.1 - Prob. 27PCh. 5.1 - Prob. 28PCh. 5.1 - Prob. 29PCh. 5.1 - Prob. 30PCh. 5.1 - Prob. 31PCh. 5.1 - Prob. 32PCh. 5.1 - Prob. 33PCh. 5.1 - Prob. 34PCh. 5.1 - Prob. 35PCh. 5.1 - Prob. 36PCh. 5.1 - Prob. 37PCh. 5.1 - Prob. 38PCh. 5.1 - Prob. 39PCh. 5.1 - Prob. 40PCh. 5.1 - Prob. 41PCh. 5.1 - Prob. 42PCh. 5.1 - Prob. 43PCh. 5.1 - Prob. 44PCh. 5.1 - Prob. 45PCh. 5.2 - Prob. 1PCh. 5.2 - Prob. 2PCh. 5.2 - Prob. 3PCh. 5.2 - Prob. 4PCh. 5.2 - Prob. 5PCh. 5.2 - Prob. 6PCh. 5.2 - Prob. 7PCh. 5.2 - Prob. 8PCh. 5.2 - Prob. 9PCh. 5.2 - Prob. 10PCh. 5.2 - Prob. 11PCh. 5.2 - Prob. 12PCh. 5.2 - Prob. 13PCh. 5.2 - Prob. 14PCh. 5.2 - Prob. 15PCh. 5.2 - Prob. 16PCh. 5.2 - Prob. 17PCh. 5.2 - Prob. 18PCh. 5.2 - Prob. 19PCh. 5.2 - Prob. 20PCh. 5.2 - Prob. 21PCh. 5.2 - Prob. 22PCh. 5.2 - Prob. 23PCh. 5.2 - Prob. 24PCh. 5.2 - Prob. 25PCh. 5.2 - Prob. 26PCh. 5.2 - Prob. 27PCh. 5.2 - Prob. 28PCh. 5.2 - Prob. 29PCh. 5.2 - Prob. 30PCh. 5.2 - Prob. 31PCh. 5.2 - Prob. 32PCh. 5.2 - Prob. 33PCh. 5.2 - Prob. 34PCh. 5.2 - Prob. 35PCh. 5.2 - Prob. 36PCh. 5.2 - Prob. 37PCh. 5.2 - Prob. 38PCh. 5.2 - Prob. 39PCh. 5.2 - Prob. 40PCh. 5.2 - Prob. 41PCh. 5.2 - Prob. 42PCh. 5.2 - Prob. 43PCh. 5.2 - Prob. 44PCh. 5.2 - Prob. 45PCh. 5.2 - Prob. 46PCh. 5.2 - Prob. 47PCh. 5.2 - Prob. 48PCh. 5.2 - Prob. 49PCh. 5.2 - Prob. 50PCh. 5.3 - Prob. 1PCh. 5.3 - Prob. 2PCh. 5.3 - Prob. 3PCh. 5.3 - Prob. 4PCh. 5.3 - Prob. 5PCh. 5.3 - Prob. 6PCh. 5.3 - Prob. 7PCh. 5.3 - Prob. 8PCh. 5.3 - Prob. 9PCh. 5.3 - Prob. 10PCh. 5.3 - Prob. 11PCh. 5.3 - Prob. 12PCh. 5.3 - Prob. 13PCh. 5.3 - Prob. 14PCh. 5.3 - Prob. 15PCh. 5.3 - Prob. 16PCh. 5.3 - Prob. 17PCh. 5.3 - Prob. 18PCh. 5.3 - Prob. 19PCh. 5.3 - Prob. 20PCh. 5.3 - Prob. 21PCh. 5.3 - Prob. 22PCh. 5.3 - Prob. 23PCh. 5.3 - Prob. 24PCh. 5.3 - Prob. 25PCh. 5.3 - Prob. 26PCh. 5.3 - Prob. 27PCh. 5.3 - Prob. 28PCh. 5.3 - Prob. 29PCh. 5.3 - Prob. 30PCh. 5.3 - Prob. 31PCh. 5.3 - Prob. 32PCh. 5.3 - Prob. 33PCh. 5.3 - Verify Eq. (53) by substituting the expressions...Ch. 5.3 - Prob. 35PCh. 5.3 - Prob. 36PCh. 5.3 - Prob. 37PCh. 5.3 - Prob. 38PCh. 5.3 - Prob. 39PCh. 5.3 - Prob. 40PCh. 5.4 - Prob. 1PCh. 5.4 - Prob. 2PCh. 5.4 - Prob. 3PCh. 5.4 - Prob. 4PCh. 5.4 - Prob. 5PCh. 5.4 - Prob. 6PCh. 5.4 - Prob. 7PCh. 5.4 - Prob. 8PCh. 5.4 - Prob. 9PCh. 5.4 - Prob. 10PCh. 5.4 - Prob. 11PCh. 5.4 - Prob. 12PCh. 5.4 - Prob. 13PCh. 5.4 - Prob. 14PCh. 5.4 - Prob. 15PCh. 5.4 - Prob. 16PCh. 5.4 - Prob. 17PCh. 5.4 - Prob. 18PCh. 5.4 - Prob. 19PCh. 5.4 - Prob. 20PCh. 5.4 - Prob. 21PCh. 5.4 - Prob. 22PCh. 5.4 - Prob. 23PCh. 5.4 - Prob. 24PCh. 5.4 - Prob. 25PCh. 5.4 - Prob. 26PCh. 5.4 - Prob. 27PCh. 5.4 - Prob. 28PCh. 5.4 - Prob. 29PCh. 5.5 - Prob. 1PCh. 5.5 - Prob. 2PCh. 5.5 - Prob. 3PCh. 5.5 - Prob. 4PCh. 5.5 - Prob. 5PCh. 5.5 - Prob. 6PCh. 5.5 - Prob. 7PCh. 5.5 - Prob. 8PCh. 5.5 - Prob. 9PCh. 5.5 - Prob. 10PCh. 5.5 - Prob. 11PCh. 5.5 - Prob. 12PCh. 5.5 - Prob. 13PCh. 5.5 - Prob. 14PCh. 5.5 - Prob. 15PCh. 5.5 - Prob. 16PCh. 5.5 - Prob. 17PCh. 5.5 - Prob. 18PCh. 5.5 - Prob. 19PCh. 5.5 - Prob. 20PCh. 5.5 - Prob. 21PCh. 5.5 - Prob. 22PCh. 5.5 - Prob. 23PCh. 5.5 - Prob. 24PCh. 5.5 - Prob. 25PCh. 5.5 - Prob. 26PCh. 5.5 - Prob. 27PCh. 5.5 - Prob. 28PCh. 5.5 - Prob. 29PCh. 5.5 - Prob. 30PCh. 5.5 - Prob. 31PCh. 5.5 - Prob. 32PCh. 5.5 - Prob. 33PCh. 5.5 - Prob. 34PCh. 5.5 - Prob. 35PCh. 5.5 - Prob. 36PCh. 5.6 - Prob. 1PCh. 5.6 - Prob. 2PCh. 5.6 - Prob. 3PCh. 5.6 - Prob. 4PCh. 5.6 - Prob. 5PCh. 5.6 - Prob. 6PCh. 5.6 - Prob. 7PCh. 5.6 - Prob. 8PCh. 5.6 - Prob. 9PCh. 5.6 - Prob. 10PCh. 5.6 - Prob. 11PCh. 5.6 - Prob. 12PCh. 5.6 - Prob. 13PCh. 5.6 - Prob. 14PCh. 5.6 - Prob. 15PCh. 5.6 - Prob. 16PCh. 5.6 - Prob. 17PCh. 5.6 - Prob. 18PCh. 5.6 - Prob. 19PCh. 5.6 - Prob. 20PCh. 5.6 - Prob. 21PCh. 5.6 - Prob. 22PCh. 5.6 - Prob. 23PCh. 5.6 - Prob. 24PCh. 5.6 - Prob. 25PCh. 5.6 - Prob. 26PCh. 5.6 - Prob. 27PCh. 5.6 - Prob. 28PCh. 5.6 - Prob. 29PCh. 5.6 - Prob. 30PCh. 5.6 - Prob. 31PCh. 5.6 - Prob. 32PCh. 5.6 - Prob. 33PCh. 5.6 - Prob. 34PCh. 5.6 - Prob. 35PCh. 5.6 - Prob. 36PCh. 5.6 - Prob. 37PCh. 5.6 - Prob. 38PCh. 5.6 - Prob. 39PCh. 5.6 - Prob. 40PCh. 5.7 - Prob. 1PCh. 5.7 - Prob. 2PCh. 5.7 - Prob. 3PCh. 5.7 - Prob. 4PCh. 5.7 - Prob. 5PCh. 5.7 - Prob. 6PCh. 5.7 - Prob. 7PCh. 5.7 - Prob. 8PCh. 5.7 - Prob. 9PCh. 5.7 - Prob. 10PCh. 5.7 - Prob. 11PCh. 5.7 - Prob. 12PCh. 5.7 - Prob. 13PCh. 5.7 - Prob. 14PCh. 5.7 - Prob. 15PCh. 5.7 - Prob. 16PCh. 5.7 - Prob. 17PCh. 5.7 - Prob. 18PCh. 5.7 - Prob. 19PCh. 5.7 - Prob. 20PCh. 5.7 - Prob. 21PCh. 5.7 - Prob. 22PCh. 5.7 - Prob. 23PCh. 5.7 - Prob. 24PCh. 5.7 - Prob. 25PCh. 5.7 - Prob. 26PCh. 5.7 - Prob. 27PCh. 5.7 - Prob. 28PCh. 5.7 - Prob. 29PCh. 5.7 - Prob. 30PCh. 5.7 - Prob. 31PCh. 5.7 - Prob. 32PCh. 5.7 - Prob. 33PCh. 5.7 - Prob. 34P
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- Question 2 Jordon currently has a shipment with the below details: The shipment has two types of products together Product 1 Dimension of each box: 33 x 33 x 25h Gross Weight 19.95kg Quantity 280 boxes total Product 2 Dimension per box 37x37x21h GW 14.23kg Quantity 520 Total of whole shipment together 4Pallets / 230 cubic foot / 6.6CBM / GW: 3,100kg Size of Pallets: 100*120 cmand height is no more than 200cm Jordon is trying to forecast this as a larger order, however the ratio of both items are a bit different. Solve the following: i. What is the individual volume of product1 and producttwo in the above shipment? ii. volume and weight of product1 if quantity is 1260 iii. volume and weight of product 2 if quantity is 1872 iv. if initial dimensions used 4 pallets to carry load, average number of pallets required for volume (1260 + 1872) Gross weight of larger volume order ( note: average pallet may weight 20kg) V.arrow_forwardConsider the two tanks shown in the figure below. Each has a capacity of 300 gallons. At time t = 0, tank 1 contains 100 gallons of a brine solution and tank 2 contains 200 gallons of a brine solution. Each tank also initially contains 50 pounds of salt. Pure water flows into tank 1, then, a well-mixed solution flows out from tank 1 into tank 2. Finally a well-mixed solution drains out of tank 2. The three flow rates indicated in the figure are each 5 gal/min. Tank 1, capacity = 300 gal. Volume of brine =100 gal. x(t) = amount of salt (Ibs.) Tank 2, capacity = 300 gal. Volume of brine = 200 gal. y(t) = amount of salt (lbs.) (a) Write a system of differential equations that describes the amount of salt, r(t), in tank 1 and the amount of salt, y(t), in tank 2. Use the variables a and y in writing your answers below. Do not use x(t) and y(t). da dt fip dt (b) Solve the system to find formulas for x(t) and y(t). Write your answers in terms of the variable t. z(t) y(t) (c) Determine the…arrow_forwardPART A, B and C.arrow_forward
- Two tanks are connected as in Figure 1.6. Tank 1 initially contains 20 pounds of salt dissolved in 100 gallons of brine. Tank 2 initially contains 150 gallons of brine in which 90 pounds of salt are dissolved. At time zero, a brine solution containing 1/2 pound of salt per gallon is added to tank I at the rate of 5 gallons per minute, Tank 1 has an output that discharges brine into tank 2 at the rate of 5 gallons per minute, and tank 2 also has an output of 5 gallons per minute. Deter- mine the amount of salt in each tank at any time. Also, determine when the concentration of salt in tank 2 is a minimum and how much salt is in the tank at that time. Hint: Solve for the amount of salt in tank 1 at time t and use this solution to help determine the amount in tank 2. S prl min: 12 Ih gel S gil tain Tauk pal minarrow_forwardPRACTICE PROBLEM 6.5 Molly works at a sea water filtration plant. She is working with two salt water storage cylinders as shown below. H Molly sets the storage cylinder pump settings as follows: The salt water is pumped from storage cylinder 1 to storage cylinder 2 at 10L per minute and it is pumped from storage cylinder 2 to storage cylinder 1 at 10 L per minute. Molly knows that storage cylinder 1 has 50 liters of salt water in it and that storage cylinder 2 has 25 liters of salt water in it. Assuming that the salt and water is mixed perfectly in each tank and that x is the amount of salt in kg in storage cylinder 1 after time (t) minutes has passed and y is the amount of salt in kg in sto age cylinder 2 after time (t) minutes has passed; If you know that x (0)=10kg y (0)=8kg Help Molly determine the quantity of salt in each storage cylinder after t minutes. x (t) = y (t)=arrow_forwardProblem 5 114. Biology With each breath, a person at rest breathes in about 0.50 L of air, 20.9% of which is 02, and exhales the same volume of air containing 16.3% O₂. In the lungs, oxygen diffuses into the blood, and is then transported throughout the body. Severe illness (altitude sickness) and even death can result if the amount of oxygen is too low. At sea level, atmospheric pressure is 1.00 atm, but at 3048 m (10,000 ft) it is reduced to 0.695 atm; the percentage of oxygen remains the same in both cases. Suppose that the temperature is 20 °C at both altitudes. What is the net number of oxygen molecules in each complete breath (a) at sea level and (b) at an altitude of 3048 m? (c) Use the results above to explain why peo- ple feel "out of breath" and must breathe more rapidly and deeply at high altitudes.arrow_forward
- QUESTION 1 A croissant shop produces two products: bear claws (B) and almond-filled croissants (C). Each bear claw requires 6 ounces of flour, 1 ounce of yeast, and 2 TS of almond paste. An almond-filled croissant requires 3 ounces of flour, 1 ounce of yeast, and 4 TS of almond paste. The company has 6600 ounces of flour, 1400 ounces of yeast, and 4800 TS of almond paste available for today's production run. Bear claw profits are 20 cents each, and almond-filled croissant profits are 30 cents each. What is the optimal daily profit? O $400 O $440 O $380 O $420 QUESTION 2 What combination of x and y will yield the optimum for this problem? Maximize $3x + $15y, subject to (1) 2x + 4y ≤ 12 and (2) 5x + 2y <10 and (3) x, y ≥ 0. O x = 0, y = 0 O x = 0, y = 3 O x = 2, y = 0 O x = 1, y = 5 O none of the abovearrow_forwardTwo tanks are connected as in Figure 1.6. Tank 1 initially contains 20 pounds of salt dissolved in 100 gallons of brine. Tank 2 initially contains 150 gallons of brine in which 90 pounds of salt are dissolved. At time zero, a brine solution containing 1/2 pound of salt per gallon is added to tank 1 at the rate of 5 gallons per minute. Tank 1 has an output that discharges brine into tank 2 at the rate of 5 gallons per minute, and tank 2 also has an output of 5 gallons per minute. Deter- mine the amount of salt in each tank at any time. Also, determine when the concentration of salt in tank 2 is a minimum and how much salt is in the tank at that time. Hint: Solve for the amount of salt in tank 1 at time t and use this solution to help determine the amount in tank 2. 5 gal/min: 1/2 Ib'gal 5 gal'min Tank 1 Tank 2 5 gal'minarrow_forwardThe following problem deals with the open three-tank system. Fresh water flows into tank 1; mixed brine flows from tank 1 into tank 2, from tank 2 into tank 3, and out of tank 3; all at the given flow rate r gallons per minute. The initial amounts x1(0)= x0 (lb), x2(0) = 0, and x3(0) = 0 of salt in the three tanks are given, as are their volumes V1, V2, and V3 (in gallons). First solve for the amounts of salt in the three tanks at time t , then determine the maximal amount of salt that tank 3 ever contains. Finally, construct a figure showing the graphs of x1 (t) , x2(t) , and x3(t). r = 30, x0 = 27, V1 = 30, V2 = 15, V3 =10arrow_forward
- Consider two interconnected tanks as shown in the figure above. Tank 1 initial contains 100L (liters) of water and 310 g of salt, while tank 2 initially contains 100 L of water and 480 g of salt. Water containing 15 g/L of salt is poured into tank1 at a rate of 1.5 L/min while the mixture flowing into tank 2 contains a salt concentration of 30 g/L of salt and is flowing at the rate of 3.5 L/min. The two connecting tubes have a flow rate of 4 L/min from tank 1 to tank 2; and of 2.5 L/min from tank 2 back to tank 1. Tank 2 is drained at the rate of 5 L/min. You may assume that the solutions in each tank are thoroughly mixed so that the concentration of the mixture leaving any tank along any of the tubes has the same concentration of salt as the tank as a whole. (This is not completely realistic, but as in real physics, we are going to work with the approximate, rather than exact description. The 'real equations of physics are often too complicated to even write down precisely, much less…arrow_forwardInitially, two large tanks A and B each hold 100 gallons of brine. The well-stirred liquid is pumped between the tanks as shown in the figure below. Use the information given in the figure to construct a mathematical model for the number of pounds of salt x1(t) and x2(t) at time t, measured in minutes, in tanks A and B, respectively.arrow_forwardpart 3 4arrow_forward
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