A new smartphone was released by a company - Edexcel - A-Level Maths Pure - Question 10 - 2020 - Paper 1

The statement indicates that the rate of change of the total number of phones sold, represented by n, with respect to time t can be expressed mathematically as:

$\frac{dn}{dt} = k n$

where k is a positive constant representing the proportionality factor. This is a standard form of a first-order linear differential equation.

To express n in terms of t, we recognize that this differential equation describes exponential growth. The general solution can be derived as follows:

1. Rewrite the equation:

   $\frac{dn}{n} = k \, dt$

2. Integrate both sides:

   $\int \frac{1}{n} \, dn = \int k \, dt$

   This yields:

   $\ln |n| = kt + C$

   where C is the constant of integration.

3. Exponentiate both sides to solve for n:

   $n = e^{kt + C}$

4. By rewriting, we can express this in a suitable form:

   $n = A e^{kt}$

   where A = e^C is a positive constant. This shows the relationship of n in terms of t.