Mathematical Analysis and Data Trends

41 questions

All types41 Multiple choice17 Numeric15 Matching5 Ordering4

During a laboratory investigation on acoustics, researchers recorded the characteristics of sound waves traveling through a gas chamber at a constant temperature. For each trial, the wave's wavelength ( $\lambda$ , measured in meters) and its frequency ( $f$ , measured in hertz) were documented in the table below.

Trial	Wavelength ( $\text{m}$ )	Frequency ( $\text{Hz}$ )
1	$1.72$	$200$
2	$0.86$	$400$
3	$0.43$	$800$
4	$0.21$	$1,600$

According to the data, as the wavelength of the sound waves decreases, the frequency of the sound waves:

decreases, because wavelength and frequency are directly proportional.

decreases, because wavelength and frequency are inversely proportional.

increases, because wavelength and frequency are inversely proportional.

increases, because wavelength and frequency are directly proportional.

Show answer & explanation

Answer: increases, because wavelength and frequency are inversely proportional.

Answer

increases, because wavelength and frequency are inversely proportional.

As the wavelength decreases from

1.72\text{ m}

0.21\text{ m}

, the frequency increases from

200\text{ Hz}

1,600\text{ Hz}

. Because one variable increases as the other decreases, and their product (

\lambda \times f \approx 343

) is constant, the relationship is an inverse proportionality. Thus, the correct answer states that the frequency increases because the variables are inversely proportional.

Step-by-Step Solution

Analyze the table to observe how the frequency values change as wavelength values decrease.

As wavelength decreases from

1.72\text{ m}

0.21\text{ m}

, the frequency increases from

200\text{ Hz}

1,600\text{ Hz}

This establishes the direction of change (the trend) for the two variables.

Determine the mathematical relationship between the variables by multiplying the corresponding values of wavelength and frequency for each trial.

For Trial 1:

1.72 \times 200 = 344

. For Trial 2:

0.86 \times 400 = 344

. For Trial 3:

0.43 \times 800 = 344

. For Trial 4:

0.21 \times 1,600 = 336

. The product is approximately constant (around

343\text{ m/s}

, which represents the speed of sound in air).

A constant product (

xy = k

) defines an inverse proportionality relationship.

Combine the observed trend and the mathematical relationship to select the correct option.

Frequency increases because wavelength and frequency are inversely proportional.

This matches the correct option by linking the correct trend with the correct mathematical definition.

Key Concept

Direct and Inverse Proportionality

Estimated Time:1m 0s

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Mathematical Analysis and Data Trends

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Step-by-Step Solution

Key Concept

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