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

[lude1]

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}?​

 

[Mark44]

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.

 

[lude1]

Why would you graph r(t) and not r'(t)?

 

[Mark44]

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.

 

[lude1]

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.

 

[Mark44]

Right.