| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Topic | Exponential Equations & Modelling |
| Type | Finding x from given y value |
| Difficulty | Moderate -0.8 This question involves straightforward substitution into an exponential model and solving a simple exponential equation using logarithms. Part (a) requires evaluating N at t=0 and t=8, then finding the difference. Part (b) requires setting N=20 and solving for t using basic logarithm rules. Both parts are routine applications of standard techniques with no problem-solving insight required, making this easier than average. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b1.06i Exponential growth/decay: in modelling context |
A treatment is used to reduce the concentration of nitrate in the water in a pond.
The concentration of nitrate in the pond water, $N$ ppm (parts per million), is modelled by the equation
$$N = 65 - 3e^{0.1t} \quad t \in \mathbb{R} \quad t \geq 0$$
where $t$ hours is the time after the treatment was applied.
Use the equation of the model to answer parts (a) and (b).
(a) Calculate the reduction in the concentration of nitrate in the pond water in the first 8 hours after the treatment was applied. [3]
For fish to survive in the pond, the concentration of nitrate in the water must be no more than 20 ppm.
(b) Calculate the minimum time, after the treatment is applied, before fish can be safely introduced into the pond.
Give your answer in hours to one decimal place. [3]
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q13 [6]}}