Questions: According to a study, the average person can listen to 6 hours of sound per day at a sound level of 86 decibels without experiencing hearing loss. A decibel is a unit for measurement of sound intensity. For each increase of 8 decibels, the exposure time must be divided by three. For example, an average person can listen to 2 hours of sound per day at a sound level of 94 decibels without experiencing hearing loss. (One overexposure may result in temporary, but probably not permanent, hearing loss). Complete parts (a) through (c) below. a. Let T=f(d) be the number of hours of safe exposure time in one day to a sound at a level of d decibels above 86 decibels. Find an equation of f. f(d)= (Use integers or fractions for any numbers in the expression.)

Solution Steps

To find the equation \( f(d) \) that represents the number of hours of safe exposure time in one day to a sound at a level of \( d \) decibels above 86 decibels, we start with the given information: at 86 decibels, the safe exposure time is 6 hours. For every increase of 8 decibels, the exposure time is divided by 3. This suggests an exponential decay relationship. We can express this relationship as \( f(d) = 6 \times \left(\frac{1}{3}\right)^{\frac{d}{8}} \).

We start with the function \( f(d) \) that represents the number of hours of safe exposure time in one day to a sound at a level of \( d \) decibels above 86 decibels. The function is defined as: \[ f(d) = 6 \times \left(\frac{1}{3}\right)^{\frac{d}{8}} \]

To find the safe exposure time when \( d = 8 \): \[ f(8) = 6 \times \left(\frac{1}{3}\right)^{\frac{8}{8}} = 6 \times \left(\frac{1}{3}\right)^{1} = 6 \times \frac{1}{3} = 2 \]

The calculation shows that at a sound level of \( 94 \) decibels (which is \( 8 \) decibels above \( 86 \)), the safe exposure time is \( 2 \) hours.

Final Answer