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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.

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

[tex]\[ p(t) = 5 \sin (880 \pi t) \][/tex]

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

The domain of the function is [tex]\( t \geq \)[/tex] [tex]$\square$[/tex]

The range of the function is [tex]$\square \leq p(t) \leq \square$[/tex]

Sagot :

GinnyAnswer
To determine the domain and range of the function [tex]\( p(t) = 5 \sin(880 \cdot t) \)[/tex] within the given context, we follow these steps:

Domain:
1. The variable [tex]\( t \)[/tex] represents time in seconds after the tuning fork is struck.
2. Time cannot be negative in this context, as it is measured from the moment the tuning fork is struck.

Therefore, the domain of the function is:
[tex]\[ t \geq 0 \][/tex]

Range:
1. The function [tex]\( p(t) = 5 \sin(880 \cdot t) \)[/tex] is a sinusoidal function that oscillates.
2. The sine function, [tex]\(\sin(\theta)\)[/tex], varies between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
3. Since it is multiplied by 5, the amplitude of [tex]\( p(t) \)[/tex] is [tex]\(5\)[/tex]. This means the minimum value of [tex]\( p(t) \)[/tex] occurs when [tex]\(\sin(880 \cdot t) = -1\)[/tex] and the maximum value occurs when [tex]\(\sin(880 \cdot t) = 1\)[/tex].

Therefore, the range of the function is:
[tex]\[ -5 \leq p(t) \leq 5 \][/tex]

Putting it all together, the answers are:
- The domain of the function is [tex]\( t \geq \)[/tex] [tex]\(0\)[/tex].
- The range of the function is [tex]\( -5 \leq p(t) \leq \)[/tex] [tex]\(5\)[/tex].

So the complete answer to the question is:
- The domain of the function is [tex]\( t \geq \)[/tex] [tex]\(0\)[/tex]
- The range of the function is [tex]\( -5 \leq p(t) \leq \)[/tex] [tex]\(5\)[/tex]
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