Use the formula [tex]d=10 \log \left(\frac{P}{P_0}\right)[/tex] to find the answer to the nearest hundredth.
A whisper has a rating of 30 decibels. A normal conversation has a rating of about 64 decibels. How many times stronger is a normal conversation?
If [tex]9^x=243[/tex], what is the value of [tex]x[/tex]?
Answer (2)
Calculate the ratio of power using the decibel formula: P 1 P 2 = 1 0 10 d 2 − d 1 = 1 0 10 64 − 30 = 1 0 3.4 .
Evaluate 1 0 3.4 to find the ratio: 1 0 3.4 ≈ 2511.89 .
Solve the exponential equation 9 x = 243 by rewriting both sides as powers of 3: 3 2 x = 3 5 .
Equate the exponents and solve for x : 2 x = 5 , so x = 2.5 . The normal conversation is approximately 2511.89 times stronger than a whisper and the value of x is 2.5. x = 2.5
Explanation
Understanding the Problem We are given the formula d = 10 lo g ( P 0 P ) , where d is the decibel rating and P is the power. We are given that a whisper has a rating of 30 decibels and a normal conversation has a rating of 64 decibels. We want to find how many times stronger a normal conversation is than a whisper.
Defining Variables Let d 1 be the decibel rating of a whisper, so d 1 = 30 . Let P 1 be the power of a whisper. Let d 2 be the decibel rating of a normal conversation, so d 2 = 64 . Let P 2 be the power of a normal conversation.
Setting up Equations We have d 1 = 10 lo g ( P 0 P 1 ) and d 2 = 10 lo g ( P 0 P 2 ) . We want to find the ratio P 1 P 2 .
Solving for Power From the equations above, we have 10 d 1 = lo g ( P 0 P 1 ) and 10 d 2 = lo g ( P 0 P 2 ) . Therefore, 1 0 10 d 1 = P 0 P 1 and 1 0 10 d 2 = P 0 P 2 .
Finding the Ratio So, P 1 = P 0 ⋅ 1 0 10 d 1 and P 2 = P 0 ⋅ 1 0 10 d 2 . Then, P 1 P 2 = P 0 ⋅ 1 0 10 d 1 P 0 ⋅ 1 0 10 d 2 = 1 0 10 d 2 − d 1 = 1 0 10 64 − 30 = 1 0 10 34 = 1 0 3.4 .
Calculating the Result Calculating 1 0 3.4 , we get approximately 2511.89. Therefore, a normal conversation is approximately 2511.89 times stronger than a whisper.
Solving for x Now, let's solve the equation 9 x = 243 . We can rewrite both sides as powers of 3: ( 3 2 ) x = 3 5 , so 3 2 x = 3 5 . Therefore, 2 x = 5 , so x = 2 5 = 2.5 .
Examples
Understanding sound intensity helps in various fields, such as acoustics, environmental science, and audio engineering. For instance, in acoustics, knowing the relative intensity of different sounds allows engineers to design better soundproofing materials. In environmental science, it helps in assessing noise pollution levels. In audio engineering, it's crucial for designing audio equipment and optimizing sound quality. Also, exponential equations are used in calculating population growth, radioactive decay, and compound interest. Understanding these concepts allows us to model and predict changes in various real-world scenarios.
A normal conversation is approximately 2511.89 times stronger than a whisper, and the value of x in the equation 9^x=243 is 2.5.
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