A tuning fork vibrates with a frequency of 440 hertz (cycles/second). When the tuning fork is struck, it produces a change in the normal air pressure in the room.

Question

Function p represents this situation, where
is the change in pressure, in pascals, relative to the normal air pressure in the room as a function of time, t, in seconds, after the tuning fork is struck.

What are the domain and the range within the context of this situation?

Answer 1

In the context of the situation described with the tuning fork vibrating and producing changes in air pressure, let's analyze the domain and range of the function $p(t)$, which represents the change in pressure in pascals over time $t$ in seconds.

Domain: The domain is the set of all possible values of $t$ for which the function $p(t)$ is defined. Since we are examining the pressure change over time after the tuning fork is struck, the time $t$ begins at 0 seconds (the moment the tuning fork is struck) and extends indefinitely. Thus, the domain can be expressed as:

• Domain: $t\ge 0$, or in interval notation: $[0,\mathrm{\infty })$.

Range: The range is the set of all possible values of $p(t)$. The change in pressure caused by the tuning fork's vibrations can fluctuate above and below the normal air pressure. The pressure changes typically oscillate around 0 pascals, as they represent deviations from the normal air pressure. The changes can be positive (increased pressure) or negative (decreased pressure).

Given that tuning forks produce sinusoidal oscillations, the range of $p(t)$ would encompass all possible values resulting from these oscillations, confined typically to a maximum and minimum amplitude. However, without specific amplitude information, we consider that pressure change might oscillate above and below zero:

• Range: All real numbers within the amplitude achieved by the tuning fork, usually expressed as $(-A,A)$ where $A$ is the maximum amplitude of pressure change (the actual values depend on the specific characteristics of the tuning fork and its environment).

In summary:

• Domain: $[0,\mathrm{\infty })$ (time in seconds after the tuning fork is struck)
• Range: $(-A,A)$ (change in pressure in pascals, where $A$ is the maximum amplitude)