Math Problem Statement

The intensity of a sound in units of dB is given by \displaystyle I(\text{dB})=10\log\left(\frac{I}{I_0}\right)where  and  are measured in units of W m-2 and the value of  is 10-12 W m-2.

The sound intensity of a particular brand of washing machine is  dB. 

What is the sound intensity  in W m-2?

Give your answer to 2 significant figures.

N.B. You may wish to input your answer in scientific notation. For example, to input a value of  you would enter 3.4 * 10 ^ -5 in the input box.

Solution

The intensity of a sound in decibels is given by the equation:

I(dB)=10log(II0)I(\text{dB}) = 10 \log\left(\frac{I}{I_0}\right)

where:

  • II is the sound intensity in W/m2\text{W/m}^2,
  • I0=1012W/m2I_0 = 10^{-12} \, \text{W/m}^2 is the reference sound intensity.

To find the intensity II from the decibel value I(dB)I(\text{dB}), we can rearrange the formula as follows:

  1. Start with the given equation: I(dB)=10log(II0)I(\text{dB}) = 10 \log\left(\frac{I}{I_0}\right)
  2. Divide both sides by 10: I(dB)10=log(II0)\frac{I(\text{dB})}{10} = \log\left(\frac{I}{I_0}\right)
  3. Convert the logarithmic equation to its exponential form: II0=10I(dB)10\frac{I}{I_0} = 10^{\frac{I(\text{dB})}{10}}
  4. Multiply both sides by I0I_0 to isolate II: I=I010I(dB)10I = I_0 \cdot 10^{\frac{I(\text{dB})}{10}}

Now, substitute I0=1012W/m2I_0 = 10^{-12} \, \text{W/m}^2 and the given decibel value for the washing machine (let's call this value I(dB)I(\text{dB})):

I=101210I(dB)10I = 10^{-12} \cdot 10^{\frac{I(\text{dB})}{10}}

Please provide the value of I(dB)I(\text{dB}) so I can calculate the intensity in W/m2\text{W/m}^2.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Algebra
Sound Intensity

Formulas

I(dB) = 10 * log(I / I_0)
I = I_0 * 10^(I(dB) / 10)

Theorems

Logarithmic and Exponential Relationships

Suitable Grade Level

Grades 10-12