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The displacement, [tex]d[/tex], in millimeters of a tuning fork as a function of time, [tex]t[/tex], in seconds can be modeled with the equation [tex]d=0.4 \sin (1760 \pi t)[/tex]. What is the maximum displacement of the tuning fork?

A. [tex]0.2 \, \text{mm}[/tex]
B. [tex]0.4 \, \text{mm}[/tex]
C. [tex]0.8 \, \text{mm}[/tex]
D. [tex]2.5 \, \text{mm}[/tex]



Answer :

GinnyAnswer
To determine the maximum displacement of the tuning fork, we start by examining the given equation for displacement:

[tex]\[ d = 0.4 \sin (1760 \pi t) \][/tex]

The sine function, \(\sin(x)\), oscillates between -1 and 1. In other words:

[tex]\[ -1 \leq \sin (1760 \pi t) \leq 1 \][/tex]

This means that the value of \(\sin(1760 \pi t)\) ranges from -1 to 1. To find the maximum displacement, we need to consider the maximum value that \(\sin(1760 \pi t)\) can attain, which is 1.

Let’s substitute this maximum value into the displacement equation:

[tex]\[ d_{\text{max}} = 0.4 \sin(1760 \pi t) \quad \text{where} \quad \sin(1760 \pi t) = 1 \][/tex]

This gives us:

[tex]\[ d_{\text{max}} = 0.4 \times 1 = 0.4 \text{ mm} \][/tex]

Thus, the maximum displacement of the tuning fork is:

[tex]\[ \boxed{0.4 \text{ mm}} \][/tex]

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