2024 RI H1 Q5

The current \(I\) (in amperes) in a certain electric circuit at time \(t\) (in hours) satisfies the equation \[I = k(1 + 3\mathrm{e}^{-\frac{3t}{4}}),\] where \(k\) is a positive constant.

The initial current for this electric circuit is \(8\) amperes.

1. Show that \(k = 2\) and explain in context what this value of \(k\) represents.[2]
2. Sketch, in context, the graph of \(I\) against \(t\).[2]
3. It is given that after \(n\) minutes, where \(n\) is a positive integer, the current drops to below \(26\%\) of its initial value. Find the smallest possible value of \(n\).[3]

The finite area, \(A\), between the curve in part (b), the \(t\)-axis and the lines \(t = 0\) and \(t = 4\) is given by \[A = p + \frac{q}{\mathrm{e}^3},\] where \(p\) and \(q\) are integers.

4. Use integration to find the values of \(p\) and \(q\).[4]

Mr Sim introduces a new technology that makes the current drops slower. The current \(J\) (in amperes) in the upgraded electric circuit at time \(t\) (in hours) satisfies the equation \[J = 2(1 + 3\mathrm{e}^{-bt}),\] where \(b\) is a positive constant.

5. Write down the condition to be satisfied by \(b\).[1]