Which of the following gives the change in altitude of the balloon?

In summary: I think that's why the answer choices are confusing. Some of them are just integrals, and some of them are derivatives.
  • #1
lude1
34
0

Homework Statement



The rate of change of the altutide of a hot air balloon is given by r(t)= t3 - 4t2 + 6 for 0 ≤ t ≤ 8. Which of the following expressions gives the change in altitude of the balloon during the time the altitude is decreasing?

a. ∫r(t)dt when t goes from 1.572 to 3.517
b. ∫r(t)dt when t goes from 0 to 8
c. ∫r(t)dt when t goes from 0 to 2.667
d. ∫r'(t)dt when t goes from 1.572 to 3.514
e. ∫r'(t)dt when t goes from 0 to 2.667

Homework Equations





The Attempt at a Solution



All I know is that "the rate of change" means the derivatve and when the altitude is decreasing, the answer should be negative.

But besides not knowing how to start this problem, I'm a little confused with all of the answer choices.

For answer d, the integral and derivative cancel each other out. If that's the case, does that mean the only thing you have to do is plug in the t values into r(t), like so?

∫r'(t)dt when t goes from 1.572 to 3.514
r(t) when t goes from 1.572 to 3.514
r(t)= t3 - 4t2 + 6
r(t)= {3.5143 - 43.5142 + 6} - {1.5723 - 41.5722 + 6}?​
 
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  • #2
lude1 said:

Homework Statement



The rate of change of the altutide of a hot air balloon is given by r(t)= t3 - 4t2 + 6 for 0 ≤ t ≤ 8. Which of the following expressions gives the change in altitude of the balloon during the time the altitude is decreasing?

a. ∫r(t)dt when t goes from 1.572 to 3.517
b. ∫r(t)dt when t goes from 0 to 8
c. ∫r(t)dt when t goes from 0 to 2.667
d. ∫r'(t)dt when t goes from 1.572 to 3.514
e. ∫r'(t)dt when t goes from 0 to 2.667

Homework Equations





The Attempt at a Solution



All I know is that "the rate of change" means the derivatve and when the altitude is decreasing, the answer should be negative.

But besides not knowing how to start this problem, I'm a little confused with all of the answer choices.

For answer d, the integral and derivative cancel each other out. If that's the case, does that mean the only thing you have to do is plug in the t values into r(t), like so?

∫r'(t)dt when t goes from 1.572 to 3.514
r(t) when t goes from 1.572 to 3.514
r(t)= t3 - 4t2 + 6
r(t)= {3.5143 - 43.5142 + 6} - {1.5723 - 41.5722 + 6}?​
Have you graphed r(t) = t3 - 4t2 + 6? As stated r(t) represents the time rate of change of altitude, so where r(t) > 0, the balloon is ascending, and where r(t) < 0, the balloon is descending.
 
  • #3
Why would you graph r(t) and not r'(t)?
 
  • #4
Let me turn the question around. Why would you want to graph r'(t)? What does it represent in this problem? Why wouldn't you want to graph r(t)? You know what it represents in this problem.
 
  • #5
I think why I was confused was, when it said "the rate of change", I instantly thought derivative. When I saw r(t), and not r'(t), I wanted to find r'(t) despite the fact the problem said r(t) WAS the rate of change.
 
  • #6
Right.
 

1. What is the definition of "change in altitude" for a balloon?

The change in altitude for a balloon refers to the difference in height from one position to another. It can be measured in units such as feet or meters.

2. How do you calculate the change in altitude of a balloon?

The change in altitude of a balloon can be calculated by subtracting the initial altitude from the final altitude. This will give you the total change in height for the balloon.

3. What factors can affect the change in altitude of a balloon?

The change in altitude of a balloon can be affected by factors such as wind speed, temperature, and air pressure. These factors can impact the buoyancy of the balloon and its ability to ascend or descend.

4. Why is it important to measure the change in altitude of a balloon?

Measuring the change in altitude of a balloon allows us to track its movement and understand its flight patterns. This information can be useful for weather forecasting, studying atmospheric conditions, and monitoring air traffic.

5. Is there a maximum limit to the change in altitude a balloon can reach?

Yes, there is a maximum limit to the change in altitude a balloon can reach. This limit is determined by the balloon's size, buoyancy, and the surrounding atmospheric conditions. Once the balloon reaches its maximum height, it will stop ascending and eventually begin to descend.

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