Here's the question:
You are attending a very loud concert. To avoid permanent ear damage, you decide to move farther from the stage. Sound intensity is given by the formula:
I = k/d squared
K= constant
D = distance from the listener to the source of sound.
Determine an expression for the decrease in sound intensity if you move x metres farther from the stage.
To determine the expression for the decrease in sound intensity if you move x meters farther from the stage, we need to consider the initial sound intensity and the new distance from the source of sound.
Let's start by establishing the initial sound intensity. According to the given formula, the sound intensity (I) is given by:
I = k / d^2
where k is a constant and d is the initial distance from the listener to the source of sound (in this case, the stage).
Now, suppose you move x meters farther from the stage. The new distance from the source of sound will be d + x. To find the new sound intensity, we substitute the new distance into the formula:
I' = k / (d + x)^2
where I' represents the new sound intensity.
To determine the decrease in sound intensity, we need to calculate the difference between the initial and new sound intensities:
Decrease in sound intensity = I - I'
Substituting the respective formulas for I and I' into the equation, we have:
Decrease in sound intensity = (k / d^2) - (k / (d + x)^2)
Simplifying further, we can combine the fractions:
Decrease in sound intensity = k * [(1 / d^2) - (1 / (d + x)^2)]
Therefore, the expression for the decrease in sound intensity if you move x meters farther from the stage is:
Decrease in sound intensity = k * [(1 / d^2) - (1 / (d + x)^2)]