Problem solving with logarithmic functions:
A new car has an interior sound level of 70 dB at 50 km h. A second car, at the
same speed, has an interior sound level that is two times more intense than that
of the new car. Calculate the sound level inside the second car.
To solve this problem, we can use the formula for sound level in terms of intensity:
L = 10 * log(I/I₀)
Where L is the sound level in decibels (dB), I is the intensity of sound, and I₀ is the reference intensity (which is typically 10^(-12) W/m²).
In this case, we are given the sound level of the new car (70 dB) and we need to find the sound level of the second car, given that its intensity is two times more intense.
Step 1: Convert the sound level of the new car to intensity.
Using the formula, we can rearrange it to solve for I:
I = I₀ * 10^(L/10)
Substituting the given values, we have:
I₁ = 10^(-12) * 10^(70/10)
Step 2: Find the intensity of the second car.
Since the second car has an intensity that is two times more intense than the new car, we multiply the intensity of the new car by 2:
I₂ = 2 * I₁
Step 3: Calculate the sound level of the second car.
Using the formula again, this time solving for L:
L₂ = 10 * log(I₂/I₀)
Substituting the values:
L₂ = 10 * log((2 * I₁) / 10^(-12))
Now, let's solve each step step-by-step:
Step 1:
Using a calculator to evaluate 10^(70/10), we get:
I₁ ≈ 10^7 W/m²
Step 2:
I₂ = 2 * I₁
I₂ ≈ 2 * 10^7 W/m²
I₂ ≈ 2 * 10^7 W/m²
I₂ ≈ 20 * 10^6 W/m²
I₂ ≈ 20 million W/m²
Step 3:
Using a calculator to evaluate log((2 * 10^7) / 10^(-12)), we get:
L₂ ≈ 10 * log(2 * 10^19)
L₂ ≈ 10 * (19.30)
L₂ ≈ 193
Therefore, the sound level inside the second car is approximately 193 dB.