Hello! Can someone please help me out for this question?
The loudness, L, of a sound in decibels can be calculated using the formula L= 10 log(I/ F) where I is the intensity of the sound in watts per square metre and F= 10^ -12 . A singer is performing to a crowd. Determine the intensity of the singers voice if the sound level is 90 dB.
This is what i have so far:
90 = 10 log [ I/F]
10^1.9543 = 10 log [I/F]
What should i do now?
* F is really a (I) with a zero underneath..but i didn't know how to type that in
90 = 10 log [ I/F]
Use the properties of logarithms to isolate I.
9 = log(I/F)
10^9 = I/F
Well, it seems like you're almost there! To solve for the intensity of the singer's voice, you need to isolate I in the equation. Let's keep going:
10^1.9543 = 10 log [I/F]
Now, let's get rid of the logarithm by taking both sides of the equation to the power of 10:
10^(10^1.9543) = [I/F]
Now, substitute F with its value (10^-12):
10^(10^1.9543) = [I/(10^-12)]
Now, multiply both sides of the equation by (10^-12) to isolate I:
10^(10^1.9543) * (10^-12) = I
Don't worry if it looks complicated with all those exponents! Just plug it into a calculator and work your magic. You'll find the value of I, which is the intensity of the singer's voice.
Remember, laughter is the best math exercise!
To solve for the intensity of the singer's voice, you can first rewrite the equation as an exponential equation using the logarithm property:
10^(90/10) = I/F
Next, simplify the left side of the equation:
10^9 = I/F
Now, substitute the value of F in the equation as F = 10^-12:
10^9 = I/(10^-12)
To simplify further, you can convert the exponent on the left side of the equation to match the denominator on the right side:
10^9 = I * 10^12
Now, divide both sides of the equation by 10^12 to isolate I:
I = 10^9 / 10^12
Lastly, you can simplify the right side of the equation:
I = 10^-3
Therefore, the intensity of the singer's voice is 10^-3 watts per square meter.
To continue solving the equation, you need to isolate the variable I (intensity of the sound). Here's what you should do next:
1. Rewrite the equation: 10^1.9543 = 10 log[I/F].
2. Remove the logarithm by converting it to exponential form. The log base 10 and the exponential function are inverse operations, so you can rewrite it as 10^(10^1.9543) = I / F.
3. Evaluate the exponent: 10^1.9543 ≈ 90.177.
4. Substitute the value into the equation: 10^90.177 = I / F.
5. Solve for I by multiplying both sides of the equation by F: I = (10^90.177) * F.
6. Substitute the value of F into the equation: I = (10^90.177) * (10^-12).
7. Simplify the equation by combining the exponents: I = 10^(90.177 - 12).
8. Calculate the value of the exponent: 90.177 - 12 ≈ 78.177.
9. Evaluate the final expression: I ≈ 10^78.177.
Therefore, the intensity of the singer's voice is approximately equal to 10^78.177 watts per square meter.