Question

A wind turbine uses the power of wind to generate electricity. The blades of the turbine make a noise that can be heard at a distance from the turbine. At a distance of d = 0 meters from the turbine, the noise level is 105 decibels. At a distance of d = 100 meters from the turbine, the noise level is 49 decibels. The noise level can be modeled by the function S given by Sd) = ab*, where Sd) is the noise level, in decibels, at a distance of d meters from the turbine. Part B (i Use the given data to find the average rate of change of the noise level, in decibels per meter, from d = 0 to d = 100 meters. Express your answer as a decimal approximation. Show the computations that lead to your ans ver. (i) Interpret the meaning of your answer from (i) in the context of the problem. (ili) Use the average rate of change found in i) to estimate the noise level, in decibels, at a distance of d = 120 meters. Show the work that leads to your answer.

Answer 1

i) The average rate of change of the noise level from d=0 to d=100 meters is −0.56 decibels per meter.

ii) The estimated noise level at d=120 meters is approximately 37.8 decibels.

To find the average rate of change of the noise level from d=0 to d=100 meters, you can use the formula for the average rate of change

Average Rate of Change= Change in S/ Change in d

​Given the data, the change in S is the difference in noise level at d=100 meters and d=0 meters, and the change in d is 100 meters.

Average Rate of Change= S(100)−S(0)/ 100

​From the given information,

S(0)=105 decibels and S(100)=49 decibels.

Average Rate of Change= 49−105/ 100

Average Rate of Change= −56/ 100

Average Rate of Change=−0.56

So, the average rate of change of the noise level from d=0 to d=100 meters is −0.56 decibels per meter.

Interpretation (ii):

This means that, on average, the noise level decreases by 0.56 decibels for every additional meter away from the turbine in the given range.

To estimate the noise level at d=120 meters using the average rate of change:

S(120)=S(100)+Average Rate of Change×Change in d

S(120)=49+(−0.56)×(120−100)

S(120)=49−0.56×20

S(120)=49−11.2

S(120)=37.8

So, the estimated noise level at d=120 meters is approximately 37.8 decibels.