An object dropped from rest from the top of a tall building on Planet X falls a distance d left parenthesis t right parenthesis equals 17 t squared feet in the first t seconds. Find the average rate of change of distance with respect to time as t changes from t 1equals3 to t 2equals6. This rate is known as the average​ velocity, or speed.
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Part 1
The average velocity as t changes from 3 to 6 seconds is 
  
enter your response here StartFraction feet Over sec EndFraction
.
​(Simplify your​ answer.)

To find the average velocity as \( t \) changes from \( t_1 = 3 \) seconds to \( t_2 = 6 \) seconds, we can use the formula for average rate of change:

\[
\text{Average Velocity} = \frac{d(t_2) - d(t_1)}{t_2 - t_1}
\]

Given the distance function:

\[
d(t) = 17t^2
\]

we first calculate \( d(3) \) and \( d(6) \):

1. **Calculate \( d(3) \)**:
   \[
   d(3) = 17(3^2) = 17(9) = 153 \text{ feet}
   \]

2. **Calculate \( d(6) \)**:
   \[
   d(6) = 17(6^2) = 17(36) = 612 \text{ feet}
   \]

Now, substitute these values into the average velocity formula:

\[
\text{Average Velocity} = \frac{612 - 153}{6 - 3} = \frac{459}{3} = 153 \text{ feet/second}
\]

Thus, the average velocity as \( t \) changes from 3 to 6 seconds is 

\[
\boxed{153 \text{ feet/second}}
\]

### Do you want more details or have any questions? Here are some related questions you might consider:

1. How does the average velocity relate to the instantaneous velocity?
2. What would be the distance covered if the object fell for 10 seconds?
3. How would the average velocity change if the distance function were different?
4. What factors could affect the distance fallen on different planets?
5. Can you explain the concept of acceleration in relation to this problem? 

**Tip:** Always check if the function you're working with is linear or quadratic, as it affects the calculations of average and instantaneous rates.

An object dropped from rest from the top of a tall building on Planet X falls a distance d(t) = 17t^2 feet in the first t seconds. Find the average rate of change of distance with respect to time as t changes from t1=3 to t2=6.

Discover how to calculate the average velocity of an object dropped from rest, modeled by the function d(t) = 17t^2, over the interval from t=3 to t=6 seconds. This problem illustrates key concepts in average velocity and quadratic functions.

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