Question

question 4 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 28 m/s, then its height in meters after t seconds is given by ( h(t) = 28t - 0.83t^2 ). find the average velocity over the given time intervals. 8, 9: 8, 8.5: 8, 8.1: 8, 8.01: 8, 8.001: question help: (\boldsymbol{\text{message instructor}}) submit question jump to answer

Explanation:

To find the average velocity over an interval \([a, b]\), we use the formula for average velocity: \(\text{Average Velocity} = \frac{h(b) - h(a)}{b - a}\), where \(h(t) = 28t - 0.83t^2\). We will calculate this for each interval:

For the interval \([8, 9]\):

Step 1: Calculate \(h(8)\) and \(h(9)\)

• \(h(8) = 28(8) - 0.83(8)^2 = 224 - 0.83(64) = 224 - 53.12 = 170.88\)
• \(h(9) = 28(9) - 0.83(9)^2 = 252 - 0.83(81) = 252 - 67.23 = 184.77\)

Step 2: Apply the average velocity formula

\(\text{Average Velocity} = \frac{h(9) - h(8)}{9 - 8} = \frac{184.77 - 170.88}{1} = 13.89\)

For the interval \([8, 8.5]\):

Step 1: Calculate \(h(8)\) and \(h(8.5)\)

• \(h(8) = 170.88\) (already calculated)
• \(h(8.5) = 28(8.5) - 0.83(8.5)^2 = 238 - 0.83(72.25) = 238 - 59.9675 = 178.0325\)

Step 2: Apply the average velocity formula

\(\text{Average Velocity} = \frac{h(8.5) - h(8)}{8.5 - 8} = \frac{178.0325 - 170.88}{0.5} = \frac{7.1525}{0.5} = 14.305\)

For the interval \([8, 8.1]\):

Step 1: Calculate \(h(8)\) and \(h(8.1)\)

• \(h(8) = 170.88\) (already calculated)
• \(h(8.1) = 28(8.1) - 0.83(8.1)^2 = 226.8 - 0.83(65.61) = 226.8 - 54.4563 = 172.3437\)

Step 2: Apply the average velocity formula

\(\text{Average Velocity} = \frac{h(8.1) - h(8)}{8.1 - 8} = \frac{172.3437 - 170.88}{0.1} = \frac{1.4637}{0.1} = 14.637\)

For the interval \([8, 8.01]\):

Step 1: Calculate \(h(8)\) and \(h(8.01)\)

• \(h(8) = 170.88\) (already calculated)
• \(h(8.01) = 28(8.01) - 0.83(8.01)^2 = 224.28 - 0.83(64.1601) = 224.28 - 53.252883 = 171.027117\)

Step 2: Apply the average velocity formula

\(\text{Average Velocity} = \frac{h(8.01) - h(8)}{8.01 - 8} = \frac{171.027117 - 170.88}{0.01} = \frac{0.147117}{0.01} = 14.7117\)

For the interval \([8, 8.001]\):

Step 1: Calculate \(h(8)\) and \(h(8.001)\)

• \(h(8) = 170.88\) (already calculated)
• \(h(8.001) = 28(8.001) - 0.83(8.001)^2\)
• \(28(8.001) = 224.028\)
• \(0.83(8.001)^2 = 0.83(64.016001) = 53.13328083\)
• \(h(8.001) = 224.028 - 53.13328083 = 170.89471917\)

Step 2: Apply the average velocity formula

\(\text{Average Velocity} = \frac{h(8.001) - h(8)}{8.001 - 8} = \frac{170.89471917 - 170.88}{0.001} = \frac{0.01471917}{0.001} = 14.71917\)

Final Answers:

• For \([8, 9]\): \(\boldsymbol{13.8900}\)
• For \([8, 8.5]\): \(\boldsymbol{14.3050}\)
• For \([8, 8.1]\): \(\boldsymbol{14.6370}\)
• For \([8, 8.01]\): \(\boldsymbol{14.7117}\)
• For \([8, 8.001]\): \(\boldsymbol{14.7192}\) (rounded to four decimal places)

Answer:

To find the average velocity over an interval \([a, b]\), we use the formula for average velocity: \(\text{Average Velocity} = \frac{h(b) - h(a)}{b - a}\), where \(h(t) = 28t - 0.83t^2\). We will calculate this for each interval:

For the interval \([8, 9]\):

Step 1: Calculate \(h(8)\) and \(h(9)\)

• \(h(8) = 28(8) - 0.83(8)^2 = 224 - 0.83(64) = 224 - 53.12 = 170.88\)
• \(h(9) = 28(9) - 0.83(9)^2 = 252 - 0.83(81) = 252 - 67.23 = 184.77\)

Step 2: Apply the average velocity formula

\(\text{Average Velocity} = \frac{h(9) - h(8)}{9 - 8} = \frac{184.77 - 170.88}{1} = 13.89\)

For the interval \([8, 8.5]\):

Step 1: Calculate \(h(8)\) and \(h(8.5)\)

• \(h(8) = 170.88\) (already calculated)
• \(h(8.5) = 28(8.5) - 0.83(8.5)^2 = 238 - 0.83(72.25) = 238 - 59.9675 = 178.0325\)

Step 2: Apply the average velocity formula

\(\text{Average Velocity} = \frac{h(8.5) - h(8)}{8.5 - 8} = \frac{178.0325 - 170.88}{0.5} = \frac{7.1525}{0.5} = 14.305\)

For the interval \([8, 8.1]\):

Step 1: Calculate \(h(8)\) and \(h(8.1)\)

• \(h(8) = 170.88\) (already calculated)
• \(h(8.1) = 28(8.1) - 0.83(8.1)^2 = 226.8 - 0.83(65.61) = 226.8 - 54.4563 = 172.3437\)

Step 2: Apply the average velocity formula

\(\text{Average Velocity} = \frac{h(8.1) - h(8)}{8.1 - 8} = \frac{172.3437 - 170.88}{0.1} = \frac{1.4637}{0.1} = 14.637\)

For the interval \([8, 8.01]\):

Step 1: Calculate \(h(8)\) and \(h(8.01)\)

• \(h(8) = 170.88\) (already calculated)
• \(h(8.01) = 28(8.01) - 0.83(8.01)^2 = 224.28 - 0.83(64.1601) = 224.28 - 53.252883 = 171.027117\)

Step 2: Apply the average velocity formula

\(\text{Average Velocity} = \frac{h(8.01) - h(8)}{8.01 - 8} = \frac{171.027117 - 170.88}{0.01} = \frac{0.147117}{0.01} = 14.7117\)

For the interval \([8, 8.001]\):

Step 1: Calculate \(h(8)\) and \(h(8.001)\)

• \(h(8) = 170.88\) (already calculated)
• \(h(8.001) = 28(8.001) - 0.83(8.001)^2\)
• \(28(8.001) = 224.028\)
• \(0.83(8.001)^2 = 0.83(64.016001) = 53.13328083\)
• \(h(8.001) = 224.028 - 53.13328083 = 170.89471917\)

Step 2: Apply the average velocity formula

\(\text{Average Velocity} = \frac{h(8.001) - h(8)}{8.001 - 8} = \frac{170.89471917 - 170.88}{0.001} = \frac{0.01471917}{0.001} = 14.71917\)

Final Answers:

• For \([8, 9]\): \(\boldsymbol{13.8900}\)
• For \([8, 8.5]\): \(\boldsymbol{14.3050}\)
• For \([8, 8.1]\): \(\boldsymbol{14.6370}\)
• For \([8, 8.01]\): \(\boldsymbol{14.7117}\)
• For \([8, 8.001]\): \(\boldsymbol{14.7192}\) (rounded to four decimal places)