A tank initially contains a solution of 14 pounds of salt in 50 gallons of water. Water with 1/5 pound of saltper gallon is added to the tank at 6 gal/min, and the resulting solution leaves at the same rate. Let Q(t)denote the 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) =t48✓Oll6esc1@2#3C$4%5&7O8O(90

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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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Newton’s law of cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e., the temperature of its surroundings). We let (t) denote time, T(t) be the temperature of the object at time (t), (T0 = T(0)) (the object’s temperature at (t = 0)), and Tomg be the constant ambient temperature. Set u(t) = T(t) - Tomg and show from Newton’s law of cooling that (u) satisfies the initial value problem (*) u’(t) = ku(t), u(0) = T0 - Tomg for a constant (k). Solve the initial value problem (*), i.e., find u(t)) (expressed in terms of (k), T0, and Tomg . Find T(t).
Let y(t) represent your bank account balance, in dollars, after t years. Suppose you start with $40000 in the account. Each year the account earns 6% interest, and you withdraw $3000. Write the differential equation modeling this situation. dy dt y(0) =
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
According to Newton's law of cooling, the temperature of a body changes at a rate proportional to the difference between the temperature of the body and the temperature of the surrounding medium. Thus, if Tm is the temperature of the medium and T = T(t) is the temperature of the body at time t, then T'= -k(T - Tm) where k is a positive constant. In this exercise, you will verify that T = Tm+ (To-Tm)e -kt is a solution to the differential equation. Here To denotes the initial temperature T(0) of the body. (a) First substitute the above expression for T into the left side of the differential equation. In other words, compute T'. Use an underscore "_" to write the subscripts on To and Im T'= (b) Next substitute the above expression for T into the right side of the differential equation. In other words, compute -k(T – Tm). - -k(T - Tm) (c) Are your answers in parts (a) and (b) equal? No Yes
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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