Exercises 8 (Exponential functions)
Evaluate the following:
- \(y=200e^{1.75}\)
- \(y=40.8+e^2\)
- \(s=3e^{5.1}-49.8\)
- \(x=(150e^{-1.34})-(3.4e^{0.445})\)
- \(x=3.2e^3 - (6-e^{-0.8})\)
Calculate the value of \(x\) in each case:
- \(25=e^x\)
- \(34=3e^x\)
- \(268=e^{1.4x}\)
- \(94.2=e^{-3.2x}\)
- \(0.72=e^{-0.08x}\)
If \(T=R\ln\left(\frac{a}{a-b}\right)\), calculate \(a\) when \(T=25\), \(R=28\) and \(b=3\).
In the formula \(i=Ie^{-\frac{Rt}{L}}\), \(i=50\,\mathrm{mA}\), \(I=150\,\mathrm{mA}\) (units: miliAmperes), \(R=60\,\Omega\) (units: Ohm) and \(L=0.3\,\mathrm{H}\) (units: Henry). Determine the corresponding value of \(t\). (Recall that for units, we have \(\mathrm{H}/\Omega = \mathrm{s}\). Optional bit: This formula describes current through the inductor when discharging in an RL-circuit.)
The instantaneous charge in a capacitive circuit is given by \(q=Q\left(1-e^{-\frac{t}{RC}}\right)\). Calculate the value of \(t\) (time) when \(q=0.01\,\mathrm{C}\), \(Q=0.015\,\mathrm{C}\) (units: Coulomb), \(C=0.0001\,\mathrm{F}\) (units: Farad) and \(R=7000\,\Omega\) (units: Ohm). (Recall that for units, we have \(\Omega\cdot\mathrm{F}=\mathrm{s}\).)
From the formula \(v=V\left(1-e^{-\frac{t}{RC}}\right)\), calculate the value of \(C\) (capacitance, units: Farad) when \(v=130\,\mathrm{V}\), \(V=440\,\mathrm{V}\) (units: Volt), \(t=0.156\,\mathrm{s}\) (units: second) and \(R=44000\Omega\) (units: Ohm). (Recall that for units, we have \(\Omega\cdot\mathrm{F}=\mathrm{s}\). Optional bit: This formula describes the instantaneous voltage over a capacitor when charging in an RC-circuit.)
Plot the function \(y = 3e^{2x}\) over the range \(x = -3\) to \(x = 3\) and from the graph determine the value of \(y\) when \(x = 1.7\) and the value of \(x\) when \(y = 3.3\).
For values of \(x\) from \(-0.5\) to \(1.5\), plot the graph represented by the equation \(y = 10e^{2x}\).
Given the formula \(i=\frac{E}{R}\left(1-e^{-\frac{Rt}{L}}\right)\), plot the curve of \(i\) against \(t\) when \(E=300\), \(R=30\) and \(L=5\) for the range of \(t\) from \(0\) to \(0.8\). From the graph, estimate the value of \(t\) when \(i=3.2\) and also calculate the value of \(t\) using the formula to check the accuracy of the graph.
The formula \(i = 2(1 - e^{-10t})\) represents the relationship between the instantaneous current \(i\) (measured in Amps) and the time \(t\) (measured in seconds) in an inductive circuit. Plot a graph of \(i\) against \(t\) taking values of \(t\) from \(0\) to \(0.3\) at intervals of \(0.05\). From the graph determine the time taken for the current to increase from \(1.0\) to \(1.6\,\mathrm{A}\) and check this value by calculation.
A coil has an inductance value of \(L=2.2\,\mathrm{H}\) and a resistance \(R=15\,\Omega\). It is connected to a voltage supply with \(E=12\,\mathrm{V}\). After connection, the current \(i\) is given by \(i=\frac{E}{R}\left(1-e^{-\frac{Rt}{L}}\right)\). Draw a graph of the current plotted against time from the moment of connection and for the first \(0.8\) seconds. From the graph establish the time it will take for the current to reach \(50\%\) of its final value and check this value by calculation.
A coil of inductance \(L\) and resistance \(R\) are connected as shown below.
The switch is moved from contact A to contact B with the result that the current \(i\) decreases according to the equation \(i=I\left(1-e^{-\frac{Rt}{L}}\right)\). Draw the graph for this decrease plotting \(i\) against \(t\) for a time of \(300\,\mathrm{ms}\) after the switch is moved. From the plot estimate the current flowing \(158\,\mathrm{ms}\) after switching.
Solutions: 1. (i) \(1151\); (ii) \(48.2\); (iii) \(442\); (iv) \(33.97\); (v) \(58.7\); 2. (i) \(3.2\); (ii) \(2.43\); (iii) \(3.99\); (iv) \(-1.4\); (v) \(4.12\); 3. \(5.1\); 4. \(5.5\,\mathrm{ms}\); 5. \(769\,\mathrm{ms}\); 6. \(0.00001\,\mathrm{F}\); 9. \(t=64\times 10^{-3}\); 10. \(91.9\times 10^{-3}\,\mathrm{s}\); 11. \(0.1\,\mathrm{s}\); 12. \(1\,\mathrm{A}\)