Model and solve problems using provided differential equations, including the logistic equation, Newton’s law of cooling and radioactive decay, with and without technology.

A cup of coffee at an initial temperature of 95°C is placed in a room maintained at 20°C. After 5 minutes, the temperature of the coffee is 70°C. Using Newton's law of cooling, calculate the temperature of the coffee after 10 minutes. Give your answer correct to the nearest degree Celsius.

A radioactive sample contains (N_0 = 8{,}000\) atoms at time (t = 0\) hours. The sample decays according to the differential equation (\frac{dN}{dt} = -0.15N\), where (N(t)\) is the number of atoms remaining after (t\) hours. (a) Solve the differential equation to find an explicit formula for (N(t)\). (b) Determine the number of atoms remaining after 10 hours, correct to the nearest whole number.

A cup of coffee at 85°C is placed in a room at 20°C. After 5 minutes, the temperature of the coffee is 60°C. Which differential equation models the temperature T (in °C) of the coffee at time t (in minutes) using Newton's law of cooling?

A cup of coffee is poured at a temperature of \(95^\circ\text{C}\) into a room where the ambient temperature is \(18^\circ\text{C}\). Newton's law of cooling states that the rate of change of temperature is proportional to the difference between the object's temperature and the ambient temperature. This can be modelled by the differential equation \(\frac{dT}{dt} = -k(T - T_a)\), where \(T\) is the temperature of the coffee in degrees Celsius at time \(t\) minutes, \(T_a\) is the ambient temperature, and \(k\) is a positive constant. (a) Show that the general solution of this differential equation can be expressed as \(T = T_a + Ce^{-kt}\), where \(C\) is a constant. (1 mark) (b) Given that the coffee cools to \(75^\circ\text{C}\) after 5 minutes, determine the value of \(k\) correct to three decimal places. (2 marks) (c) Determine the time, to the nearest minute, when the temperature of the coffee reaches \(40^\circ\text{C}\). (1 mark)

A cup of coffee cools according to Newton's law of cooling. The temperature T (in °C) of the coffee after t minutes is given by T = 20 + 60e^(-0.05t). Calculate the rate at which the coffee is cooling (in °C per minute) when t = 10 minutes.

A cup of coffee cools according to Newton's law of cooling. The temperature T °C of the coffee at time t minutes is modelled by the differential equation dT/dt = -0.08(T - 20), where 20°C is the ambient room temperature. If the coffee is initially at 95°C, calculate the temperature of the coffee after 10 minutes, correct to the nearest degree.

A cup of coffee at 95°C is placed in a room where the ambient temperature is 20°C. After 5 minutes, the temperature of the coffee is 70°C. Using Newton's law of cooling, calculate the temperature of the coffee after 10 minutes. Give your answer correct to the nearest degree Celsius.