Suppose that a 200°F freshly poured cup of coffee is placed in a large room which has an ambient temperature of 72°F and immediately forgotten about. (a) Set up an initial value problem whose solution is Y(t), the temperature of the cup of coffee (in degrees Fahren- heit) t minutes after it was poured. (Hint: we don't know the rate of proportion of the cup of coffee, so feel free to call it whatever you want.) (b) Suppose after one minute, the coffee has reached a temperature of 190°F. Use this information to determine the rate of proportion of cooling. (c) Clearly state Y(t), the function which models the temperature of the cup of coffee t minutes after it's been poured. (d) How long does it take for the temperature of the coffee to be within 5% of the ambient temperature?

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)?

1. The electrical charge on a new cell phone decreases according to the function C(t)=-0.1t -0.5t +15, where t is the time in hours following a full charge and C(t) is a measure of the charge, in charge units. What is average rate at which the electrical charge decreases from t =0 to t = 4 ? 2 Average change: {(4)- 10) -0,A(4)²– 0,5(4) +15 - [-0,1 (0)² asto) + 15] | 4- 0 4 - 3-6 -0.g b. What is the (instantaneous) rate at which the electrical charge decreases when t = 4? How long does one have until the charge is fully depleted? Hint: C(t)=0 when the charge is fully C. depleted.

1A) What are the units of the rate of change of f(t) with respect of t?        What does this rate mesuare?