Question

consider the following graph, which shows the value of a painting over a 17 - month period. the instantaneous rate of change was calculated as 60.1 at the point (8, 900). which rate of change will be the closest in value to the instantaneous rate of change at (8, 900)? the instantaneous rate of change at 0 months. the average rate of change between 8 and 16 months. the average rate of change between 0 and 8 months. the average rate of change between 0 and 16 months.

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Analyze each option

Option 1: Instantaneous rate of change at 0 months

The graph's slope at $x = 0$ is different from the slope at $x = 8$.

Option 2: Average rate of change between 8 and 16 months

We have $(x_1,y_1)=(8,900)$ and $(x_2,y_2)=(16,872.22)$. The average rate of change is $\frac{872.22 - 900}{16 - 8}=\frac{- 27.78}{8}=-3.4725$.

Option 3: Average rate of change between 0 and 8 months

We have $(x_1,y_1)=(0,62.4)$ and $(x_2,y_2)=(8,900)$. The average rate of change is $\frac{900 - 62.4}{8 - 0}=\frac{837.6}{8}=104.7$.

Option 4: Average rate of change between 0 and 16 months

We have $(x_1,y_1)=(0,62.4)$ and $(x_2,y_2)=(16,872.22)$. The average rate of change is $\frac{872.22 - 62.4}{16 - 0}=\frac{809.82}{16}=50.61375$.
The value closest to $60.1$ is $50.61375$.

Answer:

The average rate of change between 0 and 16 months.