Question

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 32 m/s, then its height in meters after t seconds is given by $h(t) = 32t - 0.83t^2$. find the average velocity over the given time intervals. 6, 7: 6, 6.5: 6, 6.1: 6, 6.01: 6, 6.001: question help: video

Explanation:

To find the average velocity over a time interval \([a, b]\), we use the formula for average velocity:

\[
\text{Average Velocity} = \frac{h(b) - h(a)}{b - a}
\]

where \(h(t) = 32t - 0.83t^2\).

For the interval \([6, 7]\):

Step 1: Calculate \(h(7)\) and \(h(6)\)

• \(h(7) = 32(7) - 0.83(7)^2 = 224 - 0.83(49) = 224 - 40.67 = 183.33\)
• \(h(6) = 32(6) - 0.83(6)^2 = 192 - 0.83(36) = 192 - 29.88 = 162.12\)

Step 2: Apply the average velocity formula

\[
\text{Average Velocity} = \frac{h(7) - h(6)}{7 - 6} = \frac{183.33 - 162.12}{1} = 21.21
\]

For the interval \([6, 6.5]\):

Step 1: Calculate \(h(6.5)\) and \(h(6)\)

• \(h(6.5) = 32(6.5) - 0.83(6.5)^2 = 208 - 0.83(42.25) = 208 - 35.0675 = 172.9325\)
• \(h(6) = 162.12\) (already calculated)

Step 2: Apply the average velocity formula

\[
\text{Average Velocity} = \frac{h(6.5) - h(6)}{6.5 - 6} = \frac{172.9325 - 162.12}{0.5} = \frac{10.8125}{0.5} = 21.625
\]

For the interval \([6, 6.1]\):

Step 1: Calculate \(h(6.1)\) and \(h(6)\)

• \(h(6.1) = 32(6.1) - 0.83(6.1)^2 = 195.2 - 0.83(37.21) = 195.2 - 30.8843 = 164.3157\)
• \(h(6) = 162.12\) (already calculated)

Step 2: Apply the average velocity formula

\[
\text{Average Velocity} = \frac{h(6.1) - h(6)}{6.1 - 6} = \frac{164.3157 - 162.12}{0.1} = \frac{2.1957}{0.1} = 21.957
\]

For the interval \([6, 6.01]\):

Step 1: Calculate \(h(6.01)\) and \(h(6)\)

• \(h(6.01) = 32(6.01) - 0.83(6.01)^2 = 192.32 - 0.83(36.1201) = 192.32 - 29.979683 = 162.340317\)
• \(h(6) = 162.12\) (already calculated)

Step 2: Apply the average velocity formula

\[
\text{Average Velocity} = \frac{h(6.01) - h(6)}{6.01 - 6} = \frac{162.340317 - 162.12}{0.01} = \frac{0.220317}{0.01} = 22.0317
\]

For the interval \([6, 6.001]\):

Step 1: Calculate \(h(6.001)\) and \(h(6)\)

• \(h(6.001) = 32(6.001) - 0.83(6.001)^2 = 192.032 - 0.83(36.012001) = 192.032 - 29.8900 = 162.142\) (approximate calculation)
• \(h(6) = 162.12\) (already calculated)

Step 2: Apply the average velocity formula

\[
\text{Average Velocity} = \frac{h(6.001) - h(6)}{6.001 - 6} = \frac{162.142 - 162.12}{0.001} = \frac{0.022}{0.001} = 22.0000
\] (Note: The precise calculation for \(h(6.001)\) is: \(h(6.001) = 32(6.001) - 0.83(6.001)^2 = 192.032 - 0.83(36.012001) = 192.032 - 29.890 = 162.142\) (more precisely, \(0.83 \times 36.012001 = 29.890\), so \(192.032 - 29.890 = 162.142\)). Then, \(\frac{162.142 - 162.12}{0.001} = \frac{0.022}{0.001} = 22.0000\))

Final Answers:

• \([6, 7]\): \(\boxed{21.2100}\)
• \([6, 6.5]\): \(\boxed{21.6250}\)
• \([6, 6.1]\): \(\boxed{21.9570}\)
• \([6, 6.01]\): \(\boxed{22.0317}\)
• \([6, 6.001]\): \(\boxed{22.0000}\)

Answer:

To find the average velocity over a time interval \([a, b]\), we use the formula for average velocity:

\[
\text{Average Velocity} = \frac{h(b) - h(a)}{b - a}
\]

where \(h(t) = 32t - 0.83t^2\).

For the interval \([6, 7]\):

Step 1: Calculate \(h(7)\) and \(h(6)\)

• \(h(7) = 32(7) - 0.83(7)^2 = 224 - 0.83(49) = 224 - 40.67 = 183.33\)
• \(h(6) = 32(6) - 0.83(6)^2 = 192 - 0.83(36) = 192 - 29.88 = 162.12\)

Step 2: Apply the average velocity formula

\[
\text{Average Velocity} = \frac{h(7) - h(6)}{7 - 6} = \frac{183.33 - 162.12}{1} = 21.21
\]

For the interval \([6, 6.5]\):

Step 1: Calculate \(h(6.5)\) and \(h(6)\)

• \(h(6.5) = 32(6.5) - 0.83(6.5)^2 = 208 - 0.83(42.25) = 208 - 35.0675 = 172.9325\)
• \(h(6) = 162.12\) (already calculated)

Step 2: Apply the average velocity formula

\[
\text{Average Velocity} = \frac{h(6.5) - h(6)}{6.5 - 6} = \frac{172.9325 - 162.12}{0.5} = \frac{10.8125}{0.5} = 21.625
\]

For the interval \([6, 6.1]\):

Step 1: Calculate \(h(6.1)\) and \(h(6)\)

• \(h(6.1) = 32(6.1) - 0.83(6.1)^2 = 195.2 - 0.83(37.21) = 195.2 - 30.8843 = 164.3157\)
• \(h(6) = 162.12\) (already calculated)

Step 2: Apply the average velocity formula

\[
\text{Average Velocity} = \frac{h(6.1) - h(6)}{6.1 - 6} = \frac{164.3157 - 162.12}{0.1} = \frac{2.1957}{0.1} = 21.957
\]

For the interval \([6, 6.01]\):

Step 1: Calculate \(h(6.01)\) and \(h(6)\)

• \(h(6.01) = 32(6.01) - 0.83(6.01)^2 = 192.32 - 0.83(36.1201) = 192.32 - 29.979683 = 162.340317\)
• \(h(6) = 162.12\) (already calculated)

Step 2: Apply the average velocity formula

\[
\text{Average Velocity} = \frac{h(6.01) - h(6)}{6.01 - 6} = \frac{162.340317 - 162.12}{0.01} = \frac{0.220317}{0.01} = 22.0317
\]

For the interval \([6, 6.001]\):

Step 1: Calculate \(h(6.001)\) and \(h(6)\)

• \(h(6.001) = 32(6.001) - 0.83(6.001)^2 = 192.032 - 0.83(36.012001) = 192.032 - 29.8900 = 162.142\) (approximate calculation)
• \(h(6) = 162.12\) (already calculated)

Step 2: Apply the average velocity formula

\[
\text{Average Velocity} = \frac{h(6.001) - h(6)}{6.001 - 6} = \frac{162.142 - 162.12}{0.001} = \frac{0.022}{0.001} = 22.0000
\] (Note: The precise calculation for \(h(6.001)\) is: \(h(6.001) = 32(6.001) - 0.83(6.001)^2 = 192.032 - 0.83(36.012001) = 192.032 - 29.890 = 162.142\) (more precisely, \(0.83 \times 36.012001 = 29.890\), so \(192.032 - 29.890 = 162.142\)). Then, \(\frac{162.142 - 162.12}{0.001} = \frac{0.022}{0.001} = 22.0000\))

Final Answers:

• \([6, 7]\): \(\boxed{21.2100}\)
• \([6, 6.5]\): \(\boxed{21.6250}\)
• \([6, 6.1]\): \(\boxed{21.9570}\)
• \([6, 6.01]\): \(\boxed{22.0317}\)
• \([6, 6.001]\): \(\boxed{22.0000}\)