Answered

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 [tex]62 \, \text{m/s}[/tex], then its height in meters after [tex]t[/tex] seconds is given by [tex]h(t) = 62t - 0.83t^2[/tex].

Find the average velocity over the given time intervals.

[tex][8, 9]: \, \square[/tex]

[tex][8, 8.5]: \, \square[/tex]

[tex][8, 8.1]: \, \square[/tex]

[tex][8, 8.01]: \, \square[/tex]

[tex][8, 8.001]: \, \square[/tex]



Answer :

GinnyAnswer
To determine the average velocity of the arrow over given time intervals, we use the formula for average velocity, which is the change in height divided by the change in time:

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

where [tex]\( h(t) = 62t - 0.83t^2 \)[/tex] is the height function. Now, let's calculate the average velocity for each of the given time intervals.

### Interval [tex]\([8, 9]\)[/tex]
1. Calculate [tex]\( h(9) \)[/tex] and [tex]\( h(8) \)[/tex]:
[tex]\[ h(9) = 62 \times 9 - 0.83 \times 9^2 \][/tex]
[tex]\[ h(8) = 62 \times 8 - 0.83 \times 8^2 \][/tex]
2. Compute the average velocity:
[tex]\[ \text{Average Velocity} = \frac{h(9) - h(8)}{9 - 8} \approx 47.89 \][/tex]

### Interval [tex]\([8, 8.5]\)[/tex]
1. Calculate [tex]\( h(8.5) \)[/tex] and [tex]\( h(8) \)[/tex]:
[tex]\[ h(8.5) = 62 \times 8.5 - 0.83 \times 8.5^2 \][/tex]
[tex]\[ h(8) = 62 \times 8 - 0.83 \times 8^2 \][/tex]
2. Compute the average velocity:
[tex]\[ \text{Average Velocity} = \frac{h(8.5) - h(8)}{8.5 - 8} \approx 48.305 \][/tex]

### Interval [tex]\([8, 8.1]\)[/tex]
1. Calculate [tex]\( h(8.1) \)[/tex] and [tex]\( h(8) \)[/tex]:
[tex]\[ h(8.1) = 62 \times 8.1 - 0.83 \times 8.1^2 \][/tex]
[tex]\[ h(8) = 62 \times 8 - 0.83 \times 8^2 \][/tex]
2. Compute the average velocity:
[tex]\[ \text{Average Velocity} = \frac{h(8.1) - h(8)}{8.1 - 8} \approx 48.637 \][/tex]

### Interval [tex]\([8, 8.01]\)[/tex]
1. Calculate [tex]\( h(8.01) \)[/tex] and [tex]\( h(8) \)[/tex]:
[tex]\[ h(8.01) = 62 \times 8.01 - 0.83 \times 8.01^2 \][/tex]
[tex]\[ h(8) = 62 \times 8 - 0.83 \times 8^2 \][/tex]
2. Compute the average velocity:
[tex]\[ \text{Average Velocity} = \frac{h(8.01) - h(8)}{8.01 - 8} \approx 48.7117 \][/tex]

### Interval [tex]\([8, 8.001]\)[/tex]
1. Calculate [tex]\( h(8.001) \)[/tex] and [tex]\( h(8) \)[/tex]:
[tex]\[ h(8.001) = 62 \times 8.001 - 0.83 \times 8.001^2 \][/tex]
[tex]\[ h(8) = 62 \times 8 - 0.83 \times 8^2 \][/tex]
2. Compute the average velocity:
[tex]\[ \text{Average Velocity} = \frac{h(8.001) - h(8)}{8.001 - 8} \approx 48.7192 \][/tex]

The average velocities for the respective intervals are:

[tex]\[ [8, 9]: \approx 47.89 \][/tex]
[tex]\[ [8, 8.5]: \approx 48.305 \][/tex]
[tex]\[ [8, 8.1]: \approx 48.637 \][/tex]
[tex]\[ [8, 8.01]: \approx 48.7117 \][/tex]
[tex]\[ [8, 8.001]: \approx 48.7192 \][/tex]