A tank with a capacity of 500 gal contains 200 gal of water with 100 lb of salt in solution. Water containing 1 lb of salt per gallon is entering the tank at a rate of 3 gal/min, and the mixture is allowed to flow out of the tank at a rate of 2 gal/min. Set up, but do not solve, the differential equation describing the rate of change in pounds of salt of the mixture before the tank overflows. Please simplify you equation and include all units (Hint: The amount of solution in the tank depends on time).
A 500 liter tank initially has 200 liters of solution with a concentration of salt of 5g/L. Salt water with a concentration of 20g/L is added at a rate of 3 L/min while the well mixed solution is allowed to leave at a rate of 2 L/min. This process will continue until the tank is filled.
Please give the differential equation, initial condition, and interval of definition which models the amount of salt in the above system.
A 900 gallon tank containing 600 gallons of salt water with a concentration of \(\frac{1}{3}\) pounds of salt for every gallon of water. Fresh water is added at a rate of 3 gal/min, while the well mixed solution is emptied out at a rate of 1 gal/min. Find an equation for \(S(t)\) which describes the amount of salt in the tank for the first 5 hours of this process (i.e. Let \(S(t)\) be the number of pounds of salt in the tank after \(t\) minutes, until the moment before the tank would start to overflow in this process).
A 500 liter tank initially has 100 liters of fresh water. Salt water with a concentration of 30g/L is added at a rate of 5 L/min while the well mixed solution is allowed to leave at a rate of 1 L/min. This process will continue until the tank is filled. Find a function which will model this system over the interval \([0, t^*]\) where \(t^*\) is the time at which the tank overflows.
Suppose a dye solution is made by adding dye with a concentration of 1 lb/gal to a 20 gallon tank of fresh water at a rate of 2 gal/min, while the well mixed solution is emptied at the same rate. Find the function \(D(t)\) which describes the amount of solution in pounds after \(t\) minutes.
A tank with a capacity of 500 gallons originally contains 200 gallons of water with 50 lbs of salt in solution. Water containing 0.5 lbs of salt per gallon then enters the tank at a rate of 4 gal/min while the well mixed solution exits the tank at a rate of 1 gal/min.
a) How long before the tank begins to overflow?
b) If \(t^*\) minutes is your answer for part (a) write down an equation which models the amount of salt, \(S(t)\) lbs, in the tank for \(0 \leq t \leq t^*\).
Suppose you buy a house for $ 300,000. You initially put $ 30,000 as a down payment. So you take out a 15-yr loan for $ 270,000 with interest compounded continuously with an annual percentage rate of of \(3 \%\). What will your monthly payments be (do not take into account any additional escrow accounts)?
Suppose you want to buy a house and the current APR for a 30-yr fixed rate mortgage is 4 %. If you can only afford to pay $ 2,000 a month, what is the largest loan you can afford?
Suppose your car costs $18000. What annual interest rate must you have if you are paying $400 a month and pay off the loan in 5 years, when the interest is compounded continuously?
Suppose you wish to buy a car which costs $24,000. The car dealership charges a 5 % annual interest rate which is compounded continuously. You set up the loan so that you are paying $ 400 a month. Let \(A(t)\) denote the amount owed after \(t\) years. Set up the appropriate initial value problem, find an explicit solution for \(A(t)\), and find how long it will take to pay off the loan.
Suppose you get a mortgage for $200,000, which is compounded continuously at an annual percentage rate of 2.5 %. Suppose that you are paying $1500 a month. How long before you have paid off your loan?
Suppose a ball weighing 0.2 kg is falling. Suppose that the drag constant for the ball is \(k= 0.0004 \quad 1/m\). Suppose that the ball is dropped, so that the initial velocity is 0.
(a) What is the terminal velocity \(v_{\infty}\) of the ball?
(b) Suppose the ball hits the ground after 8 seconds. What fraction of the terminal velocity is reached when the ball hits the ground?
(c) How far did the ball fall?
Suppose for the remaining parts that there is a parachute on the ball from the previous problem that opens after 2 seconds. The parachute increases the drag constant to \(0.04 \quad 1/m.\)
(d) What is the speed of the ball the instant the chute is opened?
(e) What is the terminal velocity of the ball with the chute opened?
(f) Does the ball accelerate or decelerate when the chute is opened (i.e is the acceleration positive, negative or zero)?
A 160 lb person has a mass of about 5 slugs. Suppose such a person jumps from a plane has a drag constant of 0.0004 1/ft. Suppose the individual jumps from a height of 10000 ft.
(a) Find the terminal velocity of the individual.
(b) How long before the individual has reached 90 % of their terminal velocity?
(c) If they do not have a parachute, how long before they hit the ground?
(d) Suppose that with a parachute the drag constant increases to 0.05 1/ft. Find the terminal velocity of the individual with a parachute.
(e) If the individual starts at 90 % of their terminal velocity from part (a) how long will the chute need to be open for the speed to be reduced to within 5 ft/sec of your answer in part (d)?
(f) If the individual needs to be within 5 ft/sec of your answer in (d) to walk away uninjured, then how long should the skydiver wait to open their chute to have the most time free falling. (For simplicity assume it takes 4 seconds for the chute to open completely and that the change in velocity is negligible during that time)?
Suppose that in 2011 approximately 140 million people are born a year while only 57 million die a year. The population in 2011 reached 7 billion people.
(a) Suppose that over time the percentage of people who are born and die a year is consistent. Find a function that expresses an estimate of the world’s population \(p\), \(t\) years after 2011.
(b) What is the doubling time for the world’s population using our model from part (a)?
(c) Suppose the world’s resources limit this growth. If the carrying capacity of the world is \(P\), give the logistic growth model for the world’s population \(p\), \(t\) years after 2011 (assume that the growth rate far away from \(p(t)=P\) is approximately the same as the growth rate in 2011).
Each year fruit flies come and devastate a gardener’s crop of blackberries. Suppose that during the Summer months the flies population can double every week.
(a) If the initial population is 200 flies at the beginning of Summer, and nothing is done, find the population of the flies at the end of Summer (13 weeks later).
(b) Suppose that 2 weeks into Summer the gardener notices the problem, and puts out a trap that kills 250 flies a week. Find the population of flies 11 weeks later.
(c) If the gardener puts out the trap one week earlier, how much would the final population of flies be reduced?
In this problem we work through the details of our model for a falling object. In this model we assume that the force due to drag is \(F_{drag} = -mkv^2\) where \(m\) is the mass of the object and \(k\) is the drag constant.
(a) Newton’s second law says that \(ma = mg - F_{drag}\) so we have \(v'= g-kv^2\). Find the stationary points of this system.
(b) We call the positive stationary point the terminal velocity \(v_{\infty}\). Briefly justify that we can rewrite the equation as \(v'= k(v_{\infty}^2-v^2)\).
(c) Use partial fractions rederive the solution \(v(t) = v_{\infty}\frac{e^{2v_{\infty}kt}-1}{e^{2v_{\infty}kt}+1}\) when we impose the initial condition \(v(0) =0\).
(d) Justify why the distance travelled when falling for \(t^*\) seconds is given by \(\int_0^{t^*} v(t)dt\).
In this problem we discuss the derivations for the logistic growth model.
(a) Suppose that the enviroment for our population has a maximum capacity of \(K\) (this is sometimes called the carrying capacity). Also suppose that so long as the population is significantly less than \(K\) that the rate of growth is \(r\). Justify why the model \(p' = rp(1-p/k)\) satisfies these assumptions.
(b) Let \(p_I\) be the initial population for a population satisfying the model described in part (a). Rederive the solution \[p(t) = \frac{Kp_Ie^{rt}}{K+p_I(e^{rt}-1)}.\]
(c) Suppose that the carrying capacity for a population is \(K\), but that the population needs to be at least \(J\) to have growth In this situation \(0<J<K\). Give an autonomous model for the population which has these properties. (HINT: You will want \(0, J, \) and \(K\) to be stationary points for our equation). You do not need to solve the equation.