The current $I$ (in amperes) in a certain electric circuit at time $t$ (in hours) satisfies the

Question

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. Show that $k = 2$ and explain in context what this value of $k$ represents.[2] Sketch, in context, the graph of $I$ against $t$.[2] 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 $x$-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. 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. Write down the condition to be satisfied by $b$.[1]

Accepted Answer

(c)346 (d)p=16, q=-8 (e)0< b $<\frac{3}{4}$

2024 RI H1 Q5

2024 RI H1 Q5