A phone company models the number of active users of a new tariff by \(N(t) = 400\times2^{\frac{t}{3}}\), where t is the time in months since launch. A riva...
Question 1 Report
A phone company models the number of active users of a new tariff by \(N(t) = 400\times2^{\frac{t}{3}}\), where t is the time in months since launch. A rival company models its tariff by \(M(t) = 400\times3^{\frac{t}{6}}\).
- Find \(N(0)\). (1)
- Find the value of \(N(3)\). (2)
- Solve the equation \(400\times2^{\frac{t}{3}} = 3200\) to find the value of t. (2)
- Find \(M(6)\) and hence state, with a reason, which tariff has more active users after 6 months. (2)
Answer Details
Exponential growth models such as \(N(t)=400\times2^{\frac{t}3}\) are evaluated by substituting the given time, and compared by evaluating both models at the same time and comparing the resulting numbers of users.
(a) At launch, \(t=0\), so \(N(0) = 400\times2^{0} = 400\times1 = 400\) users. [1 mark]
(b) At \(t=3\): \(N(3) = 400\times2^{\frac33} = 400\times2^{1} = 800\) users. [2 marks]
(c) Setting \(N(t)=3200\): \(400\times2^{\frac{t}3}=3200\), so \(2^{\frac{t}3} = \dfrac{3200}{400} = 8 = 2^{3}\). Equating the exponents, \(\dfrac{t}3=3\), so \(t=9\) months. [2 marks]
(d) The rival's model at \(t=6\): \(M(6) = 400\times3^{\frac66} = 400\times3^{1} = 1200\) users. The original tariff at \(t=6\): \(N(6) = 400\times2^{\frac63} = 400\times2^{2} = 1600\) users. Since \(1600 \gt 1200\), the first tariff has more active users after 6 months. [2 marks]
Exam tip: to compare two exponential models, evaluate both fully at the same value of \(t\) rather than trying to compare their formulas directly.
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