A new smartphone was released by a company. The company monitored the total number of phones sold, n, at time t days after the phone was released. The company observed that, during this time, the rate of increase of n was proportional to n. Use this information to write down a suitable equation for n in terms of t. (You do not need to evaluate any unknown constants in your equation.)

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A new smartphone was released by a company - Edexcel - A-Level Maths Pure - Question 9 - 2020 - Paper 1

Question 9

A new smartphone was released by a company. The company monitored the total number of phones sold, n, at time t days after the phone was released. The company obse... show full transcript

Worked Solution & Example Answer:A new smartphone was released by a company - Edexcel - A-Level Maths Pure - Question 9 - 2020 - Paper 1

Step 1

the rate of increase of n was proportional to n

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Answer

The rate of increase of a quantity n can be expressed mathematically by the differential equation:

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

where k is a positive constant representing the proportionality factor.

Step 2

Use this information to write down a suitable equation for n in terms of t

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Answer

To solve the differential equation, we can separate variables and integrate:

Separate variables: $\frac{1}{n} dn = k dt$
Integrate both sides: $\int \frac{1}{n} dn = \int k dt$ yielding $\ln(n) = kt + C$ where C is the integration constant.
Exponentiate both sides: $n = e^{kt + C}$ simplifying to $n = A e^{kt}$ where A = e^C is a positive constant.

Thus, a suitable equation for n in terms of t is: $n = A e^{kt}$

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