A tank initially contains a solution of 7 pounds of salt in 60 gallons of water. Water with 3/10 pound of salt pergallon is added to the tank at 6 gal/min, and the resulting solution leaves at the same rate. Let Q(t) denotethe quantity (lbs) of salt at time t (min).(a) Write a differential equation for Q(t).Q'(t) =(b) Find the quantity Q(t) of salt in the tank at time t > 0.(c) Compute the limit.lim Q(t) =t→∞

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Answered: A tank initially contains a solution of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A virus has already infected 4 million people in a country before a vaccine becomes available to the public. Once vaccination starts, the number of infected people decreases at a rate that is proportional to the cube of the number of infected people. One month later, there are still 2 million people that are infected. Let p be the number of infected people (in millions) and t be the number of months since vaccination has started. (a) Find a differential equation and initial conditions to model the situation. (b) Solve the differential equation to find pp as a function of t (c) How many people are still infected three months after vaccination starts?
A 200 gallon tank initially contains 100 gallons of water with 20 pounds of salt. A salt solution with 1/5 pound of salt per gallon is added to the tank at 10 gal/min, and the resulting mixture is drained out at 5 gal/min. Let Q(t) denote the quantity (lbs) of salt at time t (min). (a) Write a differential equation for Q(t) which is valid up until the point at which the tank overflows. 50t+ Q' (t) = X Check your variables you might be using an incorrect one. 50+t (b) Find the quantity of salt in the tank as it's about to overflow. 38.5 X Oll ✓ @ W # 3 e t² + c C $ 4 % 5 + 6 & 7 O * 8 O 9 11 OF
A biologist is tracking Lyme Disease cases in a deer population of 8500 in a state forest in Wisconsin. Suppose at time t = 0 the number D(t) of deer with Lyme Disease is 2250 and is increasing at a rate of 110 deer per month. Assume that D'(t) is proportional to the product of the number of deer who have caught Lyme's disease and of those who have not. (a) Write a differential equation to represent this scenario. Use k for your proportionality constant and D(t) as your function (not D). D'(t) kD(t) (8500 – D(t)) ✓ os = می (b) If you did part (a) correctly, the general solution to your differential equation is D(t) = 8500 1+ Ce-8500kt Find C (keep C exact). C = 25 9 من (c) Find k. (Hint: you do NOT need the function in part (b) to find k). Keep k exact. k 1 125000 (d) How long will it take for 50% of the deer population to develop Lyme Disease? Keep k and C exact until your final calculation. Round your final answer to two decimal places. 13.59 ✓ months
A child inflates a balloon, admires it for a while and then lets the air out at a constant rate. If V(t) gives the volume of the balloon at time t, then the figure shows V'(t) as a function of t. At what time does the child do each of the following? y'(t) 1 -2 3 (a) begin to inflate the balloon? t = (b) finish inflating the balloon? t = (c) begin to let the air out? it = 1 6 0-2 3 3 (d) What would the graph of V'(t) look like if the child had alternated between pinching and releasing the open end of the balloon, instead of letting the air out at a constant rate? '(t) V'(t) 6 9 6 9 12 15 v'(t) 9 12 15 18 12 t 18 t nna: 15 18 1 -2 3 6 6 9 v'(t) 9 12 15 18 12 18 t t
As a salt dissolves in methanol, the number x(t) of grams of the salt in a solution after t seconds satisfies the dx differential equation =1.6x-0.004x² (a) What is the maximum amount of the salt that will ever dissolve in the methanol? (b) If x=50 when t=0, how long will it take for an additional 50 g of salt to dissolve? (a) The maximum amount of the salt that will ever dissolve in the methanol is gijams. (Simplify your answer.) 5 AI 9 IN F5 F6 2 8 # B - U dt F7 → = ∞ A View an example Get more help. ↓ F8 H i 8 ↑ 2 e B F9 ↑ T' l Y F10 + + 1 8 *** F11 √i co Less F12 ● S Vi 2 Ө ÷ 1, A = Clear all L PRISC INS X U SCRLK - O HOME C V € Σ PAUSE PG UP 8 FR v -W. Check answer NUM (88) A S i x

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