C

Let’s have a coil of \(0.06\,\mathrm{H}\) inductance. The current flowing through the coil is given by \[ i=0.2\sin(100\pi t),\] where time \(t\) is measured in seconds and current \(i\) is measured in amperes. Determine the voltage induced in the coil at time \(t=2\) seconds. (Hint: Instantaneous voltage can be expressed as the derivative of current function with respect to time: \(u(t)=-L\frac{\mathrm{d}i}{\mathrm{d}t}\). The negative sign indicates only that voltage induced opposes the change in current through the coil per unit time. It does not affect the magnitude of the voltage.)

For a given object to move with uniform acceleration, the engine must perform work that is related with time by the formula \[ W=3t^2, \] where work \(W\) is measured in joules and time \(t\) is measured in seconds. Determine the instantaneous engine power at time \(t=4\,\mathrm{s}\). (Hint: Instantaneous power of a given object can be expressed as the derivative of work function with respect to time: \(P(t)=\frac{\mathrm{d}W}{\mathrm{d}t}\).)

Consider non-uniform motion of an object whose position as a function of time is given by \[ s=t^3-t^2+\frac12 t, \] where time \(t\) is measured in seconds and position \(s\) is measured in meters. Find the instantaneous acceleration of the object at time \(t = 2\) s. (Hint: Instantaneous acceleration can be expressed as the derivative of the velocity function with respect to time and since velocity is the derivative of position function, instantaneous acceleration is its second derivative: \(a(t)=\frac{\mathrm{d}v}{\mathrm{d}t}=\frac{\mathrm{d}^2s}{\mathrm{d}t^2}\).)

The patient took a single dose of \(50\ \mathrm{mg}\) of the drug. Within \(3\) hours \(40\%\) of the dose was excreted from his body. The mass \(m\) (mg) of the drug in the body after time \(t\) (hours) is given by the formula \(m(t)=m_0a^t\), where \(m_0\) (mg) is the initial mass and \(a\) is a constant. Calculate how much medicine the patient had in his body after \(12\) hours.