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 meters from the turbine, the noise level is decibels. At a distance of meters from the turbine, the noise level is decibels. The noise level can be modeled by the function given by , where is the noise level, in decibels, at a distance of 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 to meters. Express your answer as a decimal approximation. Show the computations that lead to your answer.

(ii) Interpret the meaning of your answer from (i) in the context of the problem.

(iii) Use the average rate of change found in (i) to estimate the noise level, in decibels, at a distance of meters. Show the work that leads to your answer.

Part C

Use the model to find the value of such that the noise level is decibels at a distance of meters from the turbine.

Answer 1

The average rate of change of the noise level is approximately -0.33 decibels per meter. As the distance from the turbine increases, the noise level decreases by approximately 0.33 decibels per meter. The estimated noise level at a distance of 45 meters is approximately 55.05 decibels.

(i)To find the average rate of change of the noise level, we need to calculate the difference in noise level (in decibels) divided by the difference in distance (in meters).

Using the given data:

Noise level at 30 meters = 60 decibels

Noise level at 60 meters = 50 decibels

Difference in noise level = 50 - 60 = -10 decibels

Difference in distance = 60 - 30 = 30 meters

Average rate of change = (-10 decibels) / (30 meters) = -0.33 decibels per meter

Therefore, the average rate of change of the noise level is approximately -0.33 decibels per meter.

(ii)

The negative average rate of change (-0.33 decibels per meter) means that as the distance from the turbine increases, the noise level decreases. In this case, for every meter further away from the turbine, the noise level decreases by approximately 0.33 decibels.

(iii)To estimate the noise level at 45 meters, we can use the average rate of change found in part (i).

Using the noise level at 30 meters as a reference:

Noise level at 30 meters = 60 decibels

Difference in distance = 45 - 30 = 15 meters

Estimated noise level = 60 decibels + (-0.33 decibels per meter) * 15 meters = 55.05 decibels

Therefore, the estimated noise level at a distance of 45 meters is approximately 55.05 decibels.