2)(a) When an object falls from rest at t = 0 through a resisting liquid. the rate of changedvof its velocity at time t is given by= -k(v – 600), where k is a positive constant.dt%3D(i) Show that v = 600 + Pe¬kt is a solution to the differential equation for someconstant P.(ii) If the velocity of the object at t = 3s is 25 ms-1, find P and k.(iii) Find the velocity of the object at t = 10s. Give your answer correct to onedecimal place.(iv) What is the limiting value of v as t – x?

We need to solve given differential equation and use the solution to answer other part

Answered: (a) When an object falls from rest at t…

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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dP dt =rP-ap² Think of a small population of fish growing in a public pond - P measures the number of fish and t measures the number of years since the pond was stocked. P' = 2P 0.01P² to allow harvesting of 75 fish per year, adapt the logistic model to handle this. That is, write a differential equation based on the logistic model that incorporates the idea of harvesting 75 fish per year. [Hint: (total rate) = (rate in) - (rate out) ]
The initial concentration of phosphorous (C) in a waste water pond is 107 ppm, as against the acceptable limit of 6.75x10 ppm. The concentration (C) will reduce as fresh water enters the pond. The differential equation that governs the concentration, C as a function of time (in weeks) is given by. dc +0.06 C = 0, C (0) =107 ppm dt Using Euler's method and a step size of 3.5 weeks, Calculate (.) The concentration of the phosphorus after 7 weeks. (H.) Exact concentration after 7 weeks (l.) Relative absolute error
Newton's Law of Cooling states that the rate at which the temperature of a cooling object decreases and the rate at which a warming object increases are proportional to the difference between the temperature of the object and the temperature of the surrounding medium. The differential equation formula is given as: dT = k (T – A) dt - T = temperature of object at time t А ambient temperature (aka temperature of the surrounding medium) k = constant of proportionality Okay, now here's the problem. A glass of delicious orange soda (with an initial temperature of 34°F) is placed in a room with a constant temperature of 72°F. One hour later, the temperature of the orange soda is 52°F. Find the exact simplified value of k. Hint: solve the differential equation for T and note that A is a constant.
A 90 gallon tank initially contains 10 gallons of water and 5 pounds of salt. Salt solution, at 2.5 lb/gal, begins entering the tank at a rate of 4 gal/min. Simultaneously, a 3 gal/min drain is opened. in the bottom of the tank. Let s(t) be the amount of salt in the tank at time t. Find the differential equation and initial condition for which s(t) is a solution. DO NOT SOLVE THE DIFFERENTIAL EQUATION.
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
(1) A particle has an initial displacement of s = 30 metres when t = 0. The velocity of the particle is v= 6t² - 216t+ 1944 m/s (t > 0). (a) Find the displacement of the particle at time t. (b) Find the acceleration of the particle at time t. (c) What is the acceleration and displacement when v = 0?
At t=0, a 600 liter tank initially contains 50 grams of salt dissolved in 300 liters of water. Then, water that contains 2 grams of salt per liter is poured into the tank at a rate of 5 liters per minute. The solution in the tank is constantly stirred so the salt concentration is uniform at any given time t. Water leaves the tank at a rate of 2 liters per minute. Determine when the tank will overflow and create a differential equation to determine how much salt will be in the tank after 1 hour.
A tank originally contains 120 litre of brine with a salt concentration of 4 gram of salt per litre. Then brine with 15 gram of salt per litre is poured into the tank at a rate of 6 litre/minute, and the well-stirred mixture is allowed to leave at the same rate. a. Model this situation by a differential equation with initial conditions. b. Find the concentration of salt in the tank after 20 minutes. c. Find the eventual concentration of salt in the tank. (Hint: The eventual value will be as time goes to infinity.)

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