Finding total energy, in joules, and total energy watt-hours.

[kidmo87]

First of all I'm sorry if I am in the wrong place to post this problem. I have the steps to the answers, but believe it or not, I still don't really understand. It would mean a lot if someone could help walk me through it. Thanks The power supplied by a certain battery is a constant 10 Wover the first 5 hours, zero for the following 2 hours, a value that increaseslinearly from zero to 12 W during the next 10 hours, and a power that decreaseslinearly from 12 W to zero in the following 7 hours. What is the average power,in Btu/h, during this time? Round answerto two decimal places

 

[Deleted member 4993]

First of all I'm sorry if I am in the wrong place to post this problem. I have the steps to the answers, but believe it or not, I still don't really understand. It would mean a lot if someone could help walk me through it. Thanks The power supplied by a certain battery is a constant 10 Wover the first 5 hours, zero for the following 2 hours, a value that increases linearly from zero to 12 W during the next 10 hours, and a power that decreases linearly from 12 W to zero in the following 7 hours. What is the average power,in Btu/h, during this time? Round answer
to two decimal places

Plot the piece-wise-continuous function and find the area under the curve. Involves only finding area of rectangles and triangles.

Where exactly are you stuck?

 

[kidmo87]

Thanks to you, I'm not stuck at all. Now I feel dumb for not knowing. I remember doing this in math, this is engineering class. I just forgot what this problem was called. Piecewise function! thanks.

 

[kidmo87]

Sorry, I thought I understood it, but I just cant figure out the parts that I put in the red boxes on the picture. By reading the problem, I don't understand what I have to do to write the problem as shown.

 

[HallsofIvy]

For t between 0 and 5, P is the constant 10. Graphically, that is a horizontal straight line and so the region under it is a rectangle with height 10 and width 5. What is the area of that?

For t between 5 and 7, P is the constant 0. There is no region under the graph so that area is 0.

For t between 7 and 17, P is (6/5)(t- 7). It's graph is a straight line. When t= 7, P= 0 and when t= 17, P= (6/5)(10)= 12 so the region under it is a triangle with base 17- 7= 10 and height 12. What is the area of that?

For t between 17 and 24, P is -(12/7)(t- 17)+ 12. It's graph is also a straight line. When t= 17, P= 12 and when t= 24, P= -(12/7)(7)+ 12= 0 so the region under it is a triangle with base 24- 17= 7 and height 12. What is the area of that?

The area "under the graph" from t= 0 to 24 is, of course, the sum of that.


If you want to do this the "hard" way, you could integrate:
\(\displaystyle 10\int_0^5 dt+ 0\int_5^7 dt+ \frac{6}{5}\int_7^{17} (t- 7) dt- \frac{12}{7}\int_{17}^{24} (t- 17) dt+ 12\int_{17}^{24} dt\).

 

[DrPhil]

NOTE: since you Edited the question instead of adding a post to the thread, I have no way of knowing where the insertion with the functions came from - the one you wrote red boxes on. Was that given in the problem? Please don't erase the chalkboard till everyone has had a chance to read it!

Also, since you continue to post on the "Calculus" board, I assume we can use terms like "integrate." Should we be sticking with Algebra?

The power supplied by a certain battery is a constant 10 W over the first 5 hours, zero for the following 2 hours, a value that increases linearly from zero to 12 W during the next 10 hours, and a power that decreases linearly from 12 W to zero in the following 7 hours. What is the average power, in Btu/h, during this time? Round answer to two decimal places

Starting from t=0, the problem defines to power being taken from the battery.
(1) "Over the first 5 hours" --> 0 < t < 5
(2) "following 2 hours" --> 5 < t < 7
(3) "the next 10 hours" --> 7 < t < 17
(4) "following 7 hours" --> 17 < t < 24
The power (in Watts) is thus a piecewise function. There is a linear equation for P(t) in each piece, to be found from the description of power levels in the question.

For instance, for segment (3),
"a value that increases linearly from zero to 12 W during the next 10 hours"
gives you two points on a straight line: (7,0) and (17,12). The slope of the line is (12-0)/(17-7) = 6/5, and its y-intercept is at t=7.

As to area under the function, that will have horizontal units of hours and the vertical axis will be Watts. Thus the area will be W-hr. You can find that area by integrating, OR by adding up rectangles and triangles.
(1) (10 W)(5 h) = ... W-h
(2) (0 W)(2 h) =
(3) (1/2)(12 W)(10 h) =
(4) (1/2)(12 W)(7 h) =
TOTAL = ... W-h

The average (W) is the Total W-h divided by (24 h). Finally, convert W to Btu/h
[Or, convert W-h to J, J to Btu, and divide by 24h to get Btu/h.]

 

[kidmo87]

Thanks a lot, and sorry for the confusion for editing my post like that.