question 2 of 16. step 2 of 2 correct the average home attendance per week at a class aa baseball park…

Explanation:

Step1: Differentiate $N(t)$ using the chain - rule

Let $u = 4+0.3t$, then $N(t)=950u^{\frac{1}{2}}$. The derivative of $N$ with respect to $u$ is $\frac{dN}{du}=950\times\frac{1}{2}u^{-\frac{1}{2}} = 475u^{-\frac{1}{2}}$, and the derivative of $u$ with respect to $t$ is $\frac{du}{dt}=0.3$. By the chain - rule $\frac{dN}{dt}=\frac{dN}{du}\times\frac{du}{dt}$. So $N^{\prime}(t)=475(4 + 0.3t)^{-\frac{1}{2}}\times0.3=\frac{142.5}{\sqrt{4 + 0.3t}}$.

Step2: Evaluate $N^{\prime}(t)$ at $t = 6$

Substitute $t = 6$ into $N^{\prime}(t)$: $N^{\prime}(6)=\frac{142.5}{\sqrt{4+0.3\times6}}=\frac{142.5}{\sqrt{4 + 1.8}}=\frac{142.5}{\sqrt{5.8}}\approx\frac{142.5}{2.4083}\approx59$.

Step3: Interpret the meaning

The derivative $N^{\prime}(t)$ represents the rate of change of the average home - attendance per week with respect to the number of weeks into the season. So $N^{\prime}(6)$ represents the rate of attendance is increasing by $N^{\prime}(6)$ people per week, 6 weeks into the season.

Answer:

$N^{\prime}(6)=59$ The rate of attendance is increasing by $N^{\prime}(6)$ people per week, 6 weeks into the season.