Question

data about the sound levels from a new speaker are collected at a manufacturing facility and are used to calculate the model ŷ = 61.7(0.905)^x, where x represents the distance from the speaker in meters and ŷ represents the predicted sound level in decibels. interpret the percent rate of change of the model in the context of the problem. for every increase in one meter from the speaker, the sound level is predicted to decrease by about 9.5%. for every increase in one meter from the speaker, the sound level is predicted to increase by about 90.5%. for every increase in one meter from the speaker, the sound level is predicted to increase by about 9.5%. for every increase in one meter from the speaker, the sound level is predicted to decrease by about 90.5%.

Explanation:

Step1: Analyze the exponential - decay model

The general form of an exponential - decay model is $\hat{y}=a\cdot b^{x}$, where $a$ is the initial value and $b$ is the decay factor with $0 < b<1$. In the given model $\hat{y}=61.7(0.905)^{x}$, the decay factor $b = 0.905$.

Step2: Calculate the percent - rate of change

The percent - rate of change $r$ for an exponential function $\hat{y}=a\cdot b^{x}$ is given by $r=(1 - b)\times100\%$. Substitute $b = 0.905$ into the formula: $r=(1 - 0.905)\times100\%=9.5\%$. Since $b<1$, as $x$ (distance from the speaker) increases, $\hat{y}$ (sound level) decreases.

Answer:

For every increase in one meter from the speaker, the sound level is predicted to decrease by about 9.5%.