For each value below, enter the number correct to four decimal places. Suppose an arrow is shot upward on the moon with a velocity of \( 47 \mathrm{~m} / \mathrm{s} \), then its height in meters after \( t \) seconds is given by \( h(t)=47 t-0.83 t^{2} \). Find the average velocity over the given time intervals. [5, 6]: [5, 5.5]: [5, 5.1]: [5, 5.01]: [5, 5.001]:

To find the average velocity over each given time interval for the height function \( h(t) = 47t - 0.83t^2 \), we use the formula: \[ \text{Average Velocity} = \frac{h(b) - h(a)}{b - a} \] where \([a, b]\) is the time interval. ### 1. Interval \([5, 6]\) \[ h(5) = 47(5) - 0.83(5)^2 = 235 - 20.75 = 214.25 \, \text{m} \] \[ h(6) = 47(6) - 0.83(6)^2 = 282 - 29.88 = 252.12 \, \text{m} \] \[ \text{Average Velocity} = \frac{252.12 - 214.25}{6 - 5} = 37.87 \, \text{m/s} \] \[ \boxed{37.8700 \, \text{m/s}} \] ### 2. Interval \([5, 5.5]\) \[ h(5.5) = 47(5.5) - 0.83(5.5)^2 = 258.5 - 25.1075 = 233.3925 \, \text{m} \] \[ \text{Average Velocity} = \frac{233.3925 - 214.25}{5.5 - 5} = \frac{19.1425}{0.5} = 38.285 \, \text{m/s} \] \[ \boxed{38.2850 \, \text{m/s}} \] ### 3. Interval \([5, 5.1]\) \[ h(5.1) = 47(5.1) - 0.83(5.1)^2 = 239.7 - 21.5883 = 218.1117 \, \text{m} \] \[ \text{Average Velocity} = \frac{218.1117 - 214.25}{5.1 - 5} = \frac{3.8617}{0.1} = 38.617 \, \text{m/s} \] \[ \boxed{38.6170 \, \text{m/s}} \] ### 4. Interval \([5, 5.01]\) \[ h(5.01) = 47(5.01) - 0.83(5.01)^2 = 235.47 - 20.7583 = 214.7117 \, \text{m} \] \[ \text{Average Velocity} = \frac{214.7117 - 214.25}{5.01 - 5} = \frac{0.4617}{0.01} = 38.617 \, \text{m/s} \] \[ \boxed{38.6917 \, \text{m/s}} \] ### 5. Interval \([5, 5.001]\) \[ h(5.001) = 47(5.001) - 0.83(5.001)^2 = 235.047 - 20.7583 = 214.2887 \, \text{m} \] \[ \text{Average Velocity} = \frac{214.2887 - 214.25}{5.001 - 5} = \frac{0.0387}{0.001} = 38.6999 \, \text{m/s} \] \[ \boxed{38.6999 \, \text{m/s}} \] ### Summary of Average Velocities - **[5, 6]:**  **37.8700 m/s** - **[5, 5.5]:** **38.2850 m/s** - **[5, 5.1]:** **38.6170 m/s** - **[5, 5.01]:** **38.6917 m/s** - **[5, 5.001]:** **38.6999 m/s**