You are trying to overhear a juicy conversation, but from your distance of 15.0m, it sounds like only an average whisper of 25.0 dB. So you decide to move closer to give the conversation a sound level of 70.0 dB instead.How close should you come.

You have to convert the dB to intensity and then use the relationship between intensity and distance.

So first use:
I=Io*10^(B/10dB) with Io being 10^-12
once you find the intensity for each dB (25 and 80 in this case) you use the relationship:
I1/I2=r2^2/r1^2
following the equations closely should give you the answer in meters and you just have to convert to centimeters. For this particular problem you should get around 3 cm.

sound intensity is proportional to the inverse of distance squared.

10^7/10^2.5 = 15^2/d^2

d^2= 15^2*10^-4.5

d= 15*10^-2.25= 15*.0056= 8 cm

check my thinking.

Well, if you're looking to get closer and hear that juicy gossip, let's calculate the distance you need to cover. Going from a mere whisper to a volume of 70.0 dB is quite a leap!

To determine the distance, we can use the inverse square law, which states that the sound intensity decreases as the square of the distance from the source increases. So, let's crunch the numbers!

Given:
Initial distance (d1) = 15.0 m
Initial sound level (L1) = 25.0 dB
Final sound level (L2) = 70.0 dB

Using the formula for sound level difference:

L2 - L1 = 10 * log10(I2 / I1)

Rearranging the formula to solve for intensity (I2):

I2 / I1 = 10^((L2 - L1) / 10)

Simplifying further:

(I2 / I1)^0.5 = 10^((L2 - L1) / 20)

For a tenfold increase in sound level, we need the ratio (I2 / I1) to be 10. So:

(10)^0.5 = 10^((70 - 25) / 20)

Simplifying again:

3.1623 ≈ 10^((45) / 20)

We can now solve for the unknown distance (d2):

d2 = d1 * (I1 / I2)^0.5

Plugging in the values:

d2 = 15.0 m * (1 / 3.1623)^0.5

Calculating:

d2 ≈ 5.48 m

So, to pick up that juicy conversation at a sound level of 70.0 dB, you'd need to move closer by approximately 5.48 meters! Just be sure to mind your volume when you get there or you might end up being the juiciest gossip around!

To determine how close you should come to achieve a sound level of 70.0 dB, we need to understand the concept of the Inverse Square Law.

The Inverse Square Law states that the intensity of sound decreases in proportion to the square of the distance from the sound source. Mathematically, it can be represented as:

I₁/I₂ = (D₂² / D₁²)

Where:
I₁ = Initial sound intensity (in this case, 25.0 dB)
I₂ = Final sound intensity (70.0 dB)
D₁ = Initial distance from the sound source (15.0 m)
D₂ = Final distance from the sound source (unknown)

Since sound intensity is typically measured in decibels (dB), we need to convert the decibel scale to the linear scale before applying the Inverse Square Law. The conversion formula is:

I = 10^(dB/10)

Let's calculate the initial and final sound intensities:

Initial sound intensity (I₁):
I₁ = 10^(25.0/10) = 177.83 units

Final sound intensity (I₂):
I₂ = 10^(70.0/10) = 10,000 units

Now, we can use the Inverse Square Law to find the final distance (D₂):

I₁/I₂ = (D₂² / D₁²)

Plugging in the values:

177.83/10,000 = (D₂² / 15.0²)

Simplifying the equation:

0.017783 = D₂² / 225.0

Cross-multiplying:

D₂² = 225.0 * 0.017783

D₂² ≈ 4.001

Taking the square root of both sides:

D₂ ≈ √4.001

D₂ ≈ 2.00 meters

Therefore, to achieve a sound level of 70.0 dB, you should move closer to a distance of approximately 2.00 meters from the source of the conversation.