Please can anyone answer this which I have posted previously? Anyone with knowledge of this area of maths? Thanks in anticipation. :)
Formula given:
The decibel (dB) scale for measuring loudness, d, is given by the formula
d = 10 log base ten(I X 12 to the power of 12) where I is the intensity of sound in watts per square metre.
a) Find the number of decibels of sound produced by a jet engine at a distance of 50 metres if the intensity is 10 watts per square metre.
b) Find the intensity of sound if the sound level of a pneumatic drill 10 metres away is 90 decibels.
c) Find how the value of d changes if the intensity is doubled. Give your answer to the nearest decibel.
d) Find how the value of d changes if the intensity is 10 times as great.
e) By what factor does the intensity of sound have to be multiplied in order to add 20 decibels to the sound level?
I am stuck on this as well
a) To find the number of decibels of sound produced by a jet engine at a distance of 50 meters with an intensity of 10 watts per square meter, you can use the formula given: d = 10 log base ten(I X 12 to the power of 12).
First, substitute the given values into the formula:
d = 10 log base ten(10 X 12 to the power of 12)
Next, simplify the calculation inside the parentheses:
d = 10 log base ten(10 X 12,000,000,000,000)
Then, perform the multiplication:
d = 10 log base ten(120,000,000,000,000)
Now, evaluate the logarithm using a calculator or logarithm table:
d ≈ 161.21 decibels
Therefore, the sound produced by the jet engine at a distance of 50 meters with an intensity of 10 watts per square meter is approximately 161.21 decibels.
b) To find the intensity of sound if the sound level of a pneumatic drill 10 meters away is 90 decibels, you can rearrange the formula and solve for I.
First, rewrite the formula: d = 10 log base ten(I X 12 to the power of 12)
Divide both sides of the equation by 10:
(log base ten(I X 12 to the power of 12)) = d/10
Now, convert the logarithmic equation to an exponential form:
I X 12 to the power of 12 = 10 raised to the power of (d/10)
Substitute the given value into the equation:
I X 12 to the power of 12 = 10 raised to the power of (90/10)
Evaluate the right side of the equation:
I X 12 to the power of 12 = 10 raised to the power of 9
Now, divide both sides of the equation by 12 to the power of 12:
I = (10 raised to the power of 9) / 12 to the power of 12
Calculate the values on the right side of the equation:
I ≈ 2.6381 x 10^(-10) watts per square meter
Therefore, the intensity of sound produced by the pneumatic drill 10 meters away, with a sound level of 90 decibels, is approximately 2.6381 x 10^(-10) watts per square meter.
c) To find how the value of d changes if the intensity is doubled, you can use the formula and compare the results.
First, calculate the value of d using the original intensity, I. Let's call this value d1.
d1 = 10 log base ten(I X 12 to the power of 12)
Next, substitute 2I for the intensity, I, in the formula to represent the doubled intensity. Let's call this value d2.
d2 = 10 log base ten(2I X 12 to the power of 12)
Simplify both equations separately:
d1 = 10 log base ten(I X 12 to the power of 12)
d2 = 10 log base ten(2I X 12 to the power of 12)
Now, compare the two equations, d1 and d2. Calculate the difference by subtracting d1 from d2:
d2 - d1 = 10 log base ten(2I X 12 to the power of 12) - 10 log base ten(I X 12 to the power of 12)
Using the logarithmic property, subtracting logarithms becomes division:
d2 - d1 = 10 log base ten((2I X 12 to the power of 12) / (I X 12 to the power of 12))
Simplify the equation further:
d2 - d1 = 10 log base ten(2)
Evaluate the logarithm of 2 using a calculator or logarithm table:
d2 - d1 ≈ 3.01 decibels
Therefore, doubling the intensity of sound results in a change of approximately 3.01 decibels.
d) To find how the value of d changes if the intensity is 10 times as great, you can use a similar process as in part c.
First, calculate the value of d using the original intensity, I. Let's call this value d1.
d1 = 10 log base ten(I X 12 to the power of 12)
Next, substitute 10I for the intensity, I, in the formula to represent the increased intensity. Let's call this value d2.
d2 = 10 log base ten(10I X 12 to the power of 12)
Simplify both equations separately:
d1 = 10 log base ten(I X 12 to the power of 12)
d2 = 10 log base ten(10I X 12 to the power of 12)
Now, compare the two equations, d1 and d2. Calculate the difference by subtracting d1 from d2:
d2 - d1 = 10 log base ten(10I X 12 to the power of 12) - 10 log base ten(I X 12 to the power of 12)
Using the logarithmic property, subtracting logarithms becomes division:
d2 - d1 = 10 log base ten((10I X 12 to the power of 12) / (I X 12 to the power of 12))
Simplify the equation further:
d2 - d1 = 10 log base ten(10)
Evaluate the logarithm of 10 using a calculator or logarithm table:
d2 - d1 ≈ 10 decibels
Therefore, increasing the intensity of sound by 10 times results in a change of approximately 10 decibels.
e) To find the factor by which the intensity of sound needs to be multiplied in order to add 20 decibels to the sound level, we can rearrange the formula and solve for the factor.
First, rewrite the formula: d = 10 log base ten(I X 12 to the power of 12)
Add 20 to both sides of the equation to represent the increase in decibels:
d + 20 = 10 log base ten(I X 12 to the power of 12)
Divide both sides of the equation by 10:
(log base ten(I X 12 to the power of 12)) = (d + 20)/10
Now, convert the logarithmic equation to an exponential form:
I X 12 to the power of 12 = 10 raised to the power of ((d + 20)/10)
Taking the 10th root of both sides of the equation to isolate I:
I = (10 raised to the power of ((d + 20)/10)) / 12 to the power of 12
Now, calculate the values on the right side of the equation:
I ≈ 6.3096 x 10^(-8) watts per square meter
Therefore, the intensity of sound needs to be multiplied by approximately 6.3096 x 10^(-8) in order to add 20 decibels to the sound level.