​Newton's Law of Cooling says that the rate at which a body cools is proportional to the difference C in temperature between the body and the environment around it. The temperature​ f(t) of the body at time t in hours after being introduced into an environment having constant temperature T0 is f(t)=T0+C e −kt​, where C and k are constants.   A cup of coffee with temperature 130°F is placed in a freezer with temperature 0°F. After 5 minutes, the temperature of the coffee is 87°F. Use​ Newton's Law of Cooling to find the​ coffee's temperature after 10 minutes.

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.     Find the number A(t) of grams of salt in the tank at time t.

At what speed should a fish swim upstream so as to reach its destination with the least expenditure of energy? The energy depends on the friction of the fish through the water and on the duration of the trip. If the fish swims with velocity v, the energy has been found experimentally to be proportional to (for constant k > 2) times the duration of the trip. A distance of s miles against a current of speed c requires time -c (distance divided by speed). The energy required is then proportional to k = 3, minimizing energy is equivalent to minimizing |- v³ E(v) = = V C vk s v-c. For Find the speed v with which the fish should swim in order to minimize its expenditure E(v). (Your answer will depend on c, the speed of the current.)

At 7 pm on a Wednesday evening. Mr Huxley's water tank was full. The capacity of the tank was 3 000 litres. Unfortunately, the tap on the tank was leaking in such a way that the change in volume at any time (r) hours was proportional to the volume (V) of the tank. This means that AP =-kV . dt Show that V= Ve is a solution of this equation. Given that the volume of the tank after 3 hours is 1900 litres, show that k = 0.1523 correct to 4 decimal places. (iii) By the time Mr Huxley discovered that the tank was leaking, there were only 250 litres of water remaining. At what time and on which day did Mr Huxley discover the leak (Answer correct to the nearest minute)?