If the demand for a product is given by \( D(p)=\frac{1000}{\sqrt{p}}-1 \) what is the average rate of change of demand when \( p \) increases within the following values? (Give exact answers. Do not round.) \( \begin{array}{l}\text { (a) } 4 \text { to } 16 \\ -200 / 9 \\ \text { (b) } 16 . . .16 . t o .100 \\ \text { Your answer cannot be understood or graded. More Information } \\ \text { Entaction, integer, or exact decimal. Do not approximate. }\end{array} \).

To find the average rate of change of demand between the defined price intervals, we need to calculate \( D(p) \) at both ends of each interval and then use the formula for the average rate of change: \[ \text{Average Rate of Change} = \frac{D(b) - D(a)}{b - a} \] For part (a), let's compute \( D(4) \) and \( D(16) \): 1. **Calculate \( D(4) \)**: \[ D(4) = \frac{1000}{\sqrt{4}} - 1 = \frac{1000}{2} - 1 = 500 - 1 = 499 \] 2. **Calculate \( D(16) \)**: \[ D(16) = \frac{1000}{\sqrt{16}} - 1 = \frac{1000}{4} - 1 = 250 - 1 = 249 \] Now, using these values in the formula for average rate of change: \[ \text{Average Rate of Change} = \frac{D(16) - D(4)}{16 - 4} = \frac{249 - 499}{12} = \frac{-250}{12} = \frac{-125}{6} \] For part (b), let's compute \( D(16) \) (calculated earlier) and \( D(100) \): 1. **Calculate \( D(100) \)**: \[ D(100) = \frac{1000}{\sqrt{100}} - 1 = \frac{1000}{10} - 1 = 100 - 1 = 99 \] Now, using these values in the formula for average rate of change: \[ \text{Average Rate of Change} = \frac{D(100) - D(16)}{100 - 16} = \frac{99 - 249}{84} = \frac{-150}{84} = \frac{-25}{14} \] Final Answers: (a) \(\frac{-125}{6}\), (b) \(\frac{-25}{14}\).