dT Newton's law of cooling states that the rate of change in the temperature T(t) of a body is proportional to the difference between the temperature of the medium M(t) and the temperature of the body. That is, K[M(t)- T(t)], where K is a constant. Let K=0.04 (min) and the temperature of the medium be constant, M(t) = 291 kelvins. If the body is initially at 365 kelvins, use Euler's method with h = 0.1 min to approximate the temperature of the body after (a) 30 minutes and (b) 60 minutes. (a) The temperature of the body after 30 minutes is kelvins. (Round to two decimal places as needed.) C

A rumor spreads through a school. Let y(t) be the fraction of the population that has heard the rumor at time t and assume that the rate at which the rumor spreads is proportional to the product of the fraction y of the population that has heard the rumor and the fraction 1-y that has not yet heard the rumor. That is, assume y(t) = ky(1 - y) where k is an unknown constant. The school has 1000 students in total. At 8 a.m., 97 students have heard the rumor, and by noon, half the school has heard it. Using the logistic model, determine how much time passes before 90% of the students will have heard the rumor. (Note: Since y(t) denotes a proportion, 0 ≤ y(t) ≤ 1 for all t, and y = 1 means 100% of the students know.) 90% of the students have heard the rumor after about hours.

A model for the position x(t) of an Olympic jump skier says that the acceleration a(t) = x" is proportional to the square of the velocity v(t) = x'(t). Note this model comes from taking into account the friction of air against the skier, and we are greatly simplifying the problem by not taking into account gravity. In this problem, units of distance are in meters and units of 1. time are in seconds. (a) Give an ODE in terms of v(t) only (meaning: no x or a) that corresponds to this model, and find its general solution v. (b) Recalling that our model comes from taking friction into account, which will slow down the skier, say whether you expect your proportionality constant from la to be positive or negative. (c) Use your solution v(t) from la to find x(t). (d) Use the initial conditions' x(0) = 0 and v(0) = 100 to find the unique solution for x(t), the position of the skier. (e) Assume for simplicity the following information: the skiers take 8 seconds before reaching the ground after…

A tank contains 210 liters of fluid in which 50 grams of salt is dissolved. Pure water is then pumped into the tank at a rate of 3 L/min; the well-mixed solution is pumped out at the same rate. Let the A(t) be the number of grams of salt in the tank at time t. Find the rate at which the number of grams of salt in the tank is changing at time t. dA dt Find the number A(t) of grams of salt in the tank at time t. A(t) =