Question: The figure to the right shows the total sales, in thousands of dollars, from the distribution of $x$ thousand catalogs. Find and interpret the average rate of changes of sales with respect to the number of catalogs distributed for the given changes in $x$. (a) 15 to 25 (b) 25 to 35 (c) 35 to 45 (d) What is happening to the average rate of change as the number of catalogs distributed increases? Graph: A blue increasing curve on a coordinate grid in the top-right; x-axis is “Number (in thousands)” and y-axis is “Sales (in thousands)”. The curve rises quickly at first then levels off, with marked points near $(25, 27)$, $(35, 39)$, $(45, 44)$, and $(55, 47)$. position_hint: top-right (a) The average rate of change over 15 to 25 is (Type an integer or a simplified fraction.) This number implies that from catalog distributions of 15,000 to 25,000, sales will have an average of $\$ for each additional 1000 catalogs distributed (b) The average rate of change over 25 to 35 is (Type an integer or a simplified fraction.) This number implies that from catalog distributions of 25,000 to 35,000, sales will have an average of $\$ for each additional 1000 catalogs distributed

1. **State the problem:** We need to find the average rate of change of sales with respect to the number of catalogs distributed over the intervals 15 to 25, 25 to 35, and 35 to 45 thousand catalogs. Then interpret what happens to this rate as the number of catalogs increases. 2. **Formula for average rate of change:** $$\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$$ where $f(x)$ is the sales in thousands of dollars at $x$ thousand catalogs. 3. **Given points from the graph:** - At $x=25$, $f(25) = 27$ - At $x=35$, $f(35) = 39$ - At $x=45$, $f(45) = 44$ - At $x=15$, sales are not explicitly given, but we can estimate or assume from the curve that it is approximately $15$ (since the curve rises quickly from near zero). 4. **Calculate average rate of change for (a) 15 to 25:** $$\frac{f(25) - f(15)}{25 - 15} = \frac{27 - 15}{10} = \frac{12}{10} = 1.2$$ This means sales increase by 1.2 thousand dollars for each additional 1000 catalogs distributed between 15,000 and 25,000 catalogs. 5. **Calculate average rate of change for (b) 25 to 35:** $$\frac{f(35) - f(25)}{35 - 25} = \frac{39 - 27}{10} = \frac{12}{10} = 1.2$$ This means sales increase by 1.2 thousand dollars for each additional 1000 catalogs distributed between 25,000 and 35,000 catalogs. 6. **Calculate average rate of change for (c) 35 to 45:** $$\frac{f(45) - f(35)}{45 - 35} = \frac{44 - 39}{10} = \frac{5}{10} = 0.5$$ This means sales increase by 0.5 thousand dollars for each additional 1000 catalogs distributed between 35,000 and 45,000 catalogs. 7. **Interpretation (d):** The average rate of change decreases as the number of catalogs distributed increases. Initially, sales increase rapidly with more catalogs, but the rate slows down, indicating diminishing returns on sales as more catalogs are distributed. **Final answers:** - (a) Average rate of change = $1.2$ - (b) Average rate of change = $1.2$ - (c) Average rate of change = $0.5$ - (d) The average rate of change decreases as $x$ increases, showing sales growth slows down with more catalogs.