A student measures the frequency (f, in hertz) and wavelength (\lambda, in meters) of soun… — ACT

A student measures the frequency ( $f$ , in hertz) and wavelength ( $\lambda$ , in meters) of sound waves propagating through a room at a constant temperature. The results are recorded in the table below:

Frequency ( $Hz$ )	Wavelength ( $m$ )
170	2.00
340	1.00
680	0.50
1360	0.25

Based on these results, which of the following statements best describes the relationship between the frequency and wavelength of the sound waves?

A
Wavelength is directly proportional to frequency because as frequency increases, wavelength increases.
Wavelength is inversely proportional to frequency because as frequency increases, wavelength decreases.Answer
C
Wavelength is independent of frequency because wavelength remains constant as frequency increases.
D
Wavelength is directly proportional to frequency because the product of wavelength and frequency is constant.

Answer

Wavelength is inversely proportional to frequency because as frequency increases, wavelength decreases.

The correct answer correctly identifies that wavelength and frequency have an inverse relationship. As the frequency increases, the wavelength decreases proportionally. For example, doubling the frequency from

170\text{ Hz}

340\text{ Hz}

results in the wavelength being halved from

2.00\text{ m}

1.00\text{ m}

, satisfying the mathematical definition of inverse proportionality where the product of the two variables remains constant (

170 \times 2.00 = 340 \times 1.00 = 340

Step-by-Step Solution

Analyze the trends of both variables in the provided data table.

As frequency increases from

170\text{ Hz}

1360\text{ Hz}

, the wavelength decreases from

2.00\text{ m}

0.25\text{ m}

Identifying whether variables move in the same or opposite directions is the first step in determining proportionality.

Determine the mathematical factor by which the variables change.

When frequency is doubled (from

170\text{ Hz}

340\text{ Hz}

), wavelength is halved (from

2.00\text{ m}

1.00\text{ m}

). When frequency is quadrupled (to

680\text{ Hz}

), wavelength is divided by four (to

0.50\text{ m}

This reciprocal relationship (

x \rightarrow 2x \Rightarrow y \rightarrow \frac{1}{2}y

) confirms that the two variables are inversely proportional.

Select the statement that matches this mathematical relationship.

The statement describing the relationship as inversely proportional because wavelength decreases as frequency increases is correct.

This aligns with the observed behavior in the data.

Key Concept

Inverse proportionality describes a relationship where one variable increases in proportion to the decrease in another variable, such that their product remains constant.

Estimated Time:45s

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