Answer to Question #160536 in Calculus for Jai

Question

1. The electromotive force for an electric circuit with a simplified generator

is given by 𝑓(𝑡) = 40 sin 150 𝜋𝑡 where 𝑓(𝑡) Volts is present at 𝑡 seconds.

(a) Identify the amplitude, period, and frequency of the function.

(b) Find the maximum and minimum voltages

Answer 1

• The amplitude of a $\sin x$ function is $1$ and so as $f(t)=40\sin(150\pi t)$, the amplitude is 40 (the expression in the argument of a sin function does not affect it's amplitude). The period of a $\sin x$ function is $2 \pi$, we will use it to find the period of $f(t)$ : $\sin(150\pi t+2\pi)=\sin(150\pi t)$, now by factoring the expressions in the parentheses we find $f(t+\frac{1}{75})=f(t)$ and so the period is $T=\frac{1}{75}$. The frequency is $\nu = \frac{1}{T}=75$.
• The maximum and minimum of a $\sin x$ function are $-1, 1$ and thus the minimum voltages are $40V, -40V$ (and they are attained at, for example, $t_+=\frac{1}{300}, t_-=\frac{1}{100}$). The minimum voltage in the sense of an absolute value is $0V$ (and it is attained at, for example $t_0=0$).