Answer:
![\displaystyle average=\frac{1}{2(e-1)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20average%3D%5Cfrac%7B1%7D%7B2%28e-1%29%7D)
Step-by-step explanation:
<u>Average Value of a Function</u>
Given a function g(x), we can compute the average value of g in a given interval (a,b) with the equation:
![\displaystyle average=\frac{1}{b-a}\int_{a}^{b} g(x)dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20average%3D%5Cfrac%7B1%7D%7Bb-a%7D%5Cint_%7Ba%7D%5E%7Bb%7D%20g%28x%29dx)
We use the given data
![\displaystyle average=\frac{1}{e-1}\int_{1}^{e} \frac{lnx}{x}dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20average%3D%5Cfrac%7B1%7D%7Be-1%7D%5Cint_%7B1%7D%5E%7Be%7D%20%5Cfrac%7Blnx%7D%7Bx%7Ddx)
We now compute the indefinite integral with a u-substitution
![\displaystyle I=\int \frac{lnx}{x}dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20I%3D%5Cint%20%5Cfrac%7Blnx%7D%7Bx%7Ddx)
We'll use the substitution u=lnx, du=dx/x. Then
![\displaystyle I=\int u.du](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20I%3D%5Cint%20u.du)
Integrating
![\displaystyle I=\frac{u^2}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20I%3D%5Cfrac%7Bu%5E2%7D%7B2%7D)
Since u=lnx
![\displaystyle I=\frac{ln^2x}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20I%3D%5Cfrac%7Bln%5E2x%7D%7B2%7D)
The average value is
![\displaystyle average=\frac{1}{e-1}\left|\frac{ln^2x}{2} \right|_1^e](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20average%3D%5Cfrac%7B1%7D%7Be-1%7D%5Cleft%7C%5Cfrac%7Bln%5E2x%7D%7B2%7D%20%5Cright%7C_1%5Ee)
![\displaystyle average=\frac{1}{e-1}\left(\frac{ln^2e}{2}-\frac{ln^21}{2} \right )](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20average%3D%5Cfrac%7B1%7D%7Be-1%7D%5Cleft%28%5Cfrac%7Bln%5E2e%7D%7B2%7D-%5Cfrac%7Bln%5E21%7D%7B2%7D%20%5Cright%20%29)
Since lne=1, and ln1=0
![\displaystyle average=\frac{1}{e-1}\left(\frac{1}{2}-0 \right )](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20average%3D%5Cfrac%7B1%7D%7Be-1%7D%5Cleft%28%5Cfrac%7B1%7D%7B2%7D-0%20%5Cright%20%29)
![\displaystyle average=\frac{1}{2(e-1)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20average%3D%5Cfrac%7B1%7D%7B2%28e-1%29%7D)
Answer:
18.75
Step-by-step explanation:
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A simple way to find out:
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Answer: They will take 3 3/4 hours to paint the room together.
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A more detailed explanation:
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Alice:
6 hours = 1 room
1 hour = 1/6 of the room
Tom:
10 hours = 1 room
1 hour = 1/10 of the room
Alice and Tom together:
1 hour = 1/6 + 1/10 = 4/15 of the room
4/15 of the room = 1 hour
1/15 of the room = 1 ÷ 4 = 1/4 hour
15/15 of the room = 1/4 x 15 = 15/4 = 3 3/4 hours
Answer: They will take 3 3/4 hours to paint the room together.
Answer:
22
Step-by-step explanation:
We subtract their goal by their current amount (1000-403) to get 597. We know the class needs at least 597 more cans. We divide 597 (the total needed) by 28 (students) to get 21.3... cans per student. Since each student brings an equal number of cans, we must bring in 22 cans each, leaving us with 19 extra cans.
Answer:
between 13 and 19 comes 16
after 19 comes 22
Step-by-step explanation:
they are all added by 3
1+3=4
4+3=7
7+3=10
10+3=13
13+3=16
16+3=19
and
19+3=22