30/80 = 0.375(100) = 37.5%
30 is 37.5% of 80
24 cubic meters because 24*24*24=13,824.
The x-coordinate of the midpoint will be the average of the two given x-coordinates, and the y-coordinate will be the average of the two given y-coordinates.
This can be written as
midpoint =

,
where the given points are (x1, y1) and x2, y2).
In the problem, you have (4, -2) and (-8, 6).
The midpoint will be
Answer:
Amount invested at 8 % rate = x = $ 15000
Amount invested at 9 % rate = 34000 - x = 34000 - 15000 = $ 19000
Step-by-step explanation:
Total Amount = $ 34000
Let amount invested at 8 % rate = x
Amount invested at 9 % rate = $ 34000 - x
Total interest = $ 2910

291000 = 8 x + 306000 - 9 x
x = 306000 - 291000
x = 15000
So amount invested at 8 % rate = x = $ 15000
Amount invested at 9 % rate = 34000 - x = 34000 - 15000 = $ 19000
Answer:
We verified that ![a^3+b^3+c^3-3abc=\frac{a+b+c}{2}[(a-b)^2+(b-c)^2+(c-a)^2]](https://tex.z-dn.net/?f=a%5E3%2Bb%5E3%2Bc%5E3-3abc%3D%5Cfrac%7Ba%2Bb%2Bc%7D%7B2%7D%5B%28a-b%29%5E2%2B%28b-c%29%5E2%2B%28c-a%29%5E2%5D)
Hence proved
Step-by-step explanation:
Given equation is ![a^3+b^3+c^3-3abc=\frac{a+b+c}{2}[(a-b)^2+(b-c)^2+(c-a)^2]](https://tex.z-dn.net/?f=a%5E3%2Bb%5E3%2Bc%5E3-3abc%3D%5Cfrac%7Ba%2Bb%2Bc%7D%7B2%7D%5B%28a-b%29%5E2%2B%28b-c%29%5E2%2B%28c-a%29%5E2%5D)
We have to prove that ![a^3+b^3+c^3-3abc=\frac{a+b+c}{2}[(a-b)^2+(b-c)^2+(c-a)^2]](https://tex.z-dn.net/?f=a%5E3%2Bb%5E3%2Bc%5E3-3abc%3D%5Cfrac%7Ba%2Bb%2Bc%7D%7B2%7D%5B%28a-b%29%5E2%2B%28b-c%29%5E2%2B%28c-a%29%5E2%5D)
That is to prove that LHS=RHS
Now taking RHS
![\frac{a+b+c}{2}[(a-b)^2+(b-c)^2+(c-a)^2]](https://tex.z-dn.net/?f=%5Cfrac%7Ba%2Bb%2Bc%7D%7B2%7D%5B%28a-b%29%5E2%2B%28b-c%29%5E2%2B%28c-a%29%5E2%5D)
(using
)
(adding the like terms)
![=\frac{a+b+c}{2}[2a^2+2b^2+2c^2-2ab-2bc-2ac]](https://tex.z-dn.net/?f=%3D%5Cfrac%7Ba%2Bb%2Bc%7D%7B2%7D%5B2a%5E2%2B2b%5E2%2B2c%5E2-2ab-2bc-2ac%5D)
![=\frac{a+b+c}{2}\times 2[a^2+b^2+c^2-ab-bc-ac]](https://tex.z-dn.net/?f=%3D%5Cfrac%7Ba%2Bb%2Bc%7D%7B2%7D%5Ctimes%202%5Ba%5E2%2Bb%5E2%2Bc%5E2-ab-bc-ac%5D)
![=a+b+c[a^2+b^2+c^2-ab-bc-ac]](https://tex.z-dn.net/?f=%3Da%2Bb%2Bc%5Ba%5E2%2Bb%5E2%2Bc%5E2-ab-bc-ac%5D)
Now multiply the each term to another each term in the factor
![=a^3+ab^2+ac^2-a^2b-abc-a^2c+ba62+b^3+bc^2-ab^2-b^2c-abc+ca^2+cb^2+c^3-abc-bc^2-ac^2]](https://tex.z-dn.net/?f=%3Da%5E3%2Bab%5E2%2Bac%5E2-a%5E2b-abc-a%5E2c%2Bba62%2Bb%5E3%2Bbc%5E2-ab%5E2-b%5E2c-abc%2Bca%5E2%2Bcb%5E2%2Bc%5E3-abc-bc%5E2-ac%5E2%5D)
(adding the like terms and other terms getting cancelled)
=LHS
Therefore LHS=RHS
Therefore ![a^3+b^3+c^3-3abc=\frac{a+b+c}{2}[(a-b)^2+(b-c)^2+(c-a)^2]](https://tex.z-dn.net/?f=a%5E3%2Bb%5E3%2Bc%5E3-3abc%3D%5Cfrac%7Ba%2Bb%2Bc%7D%7B2%7D%5B%28a-b%29%5E2%2B%28b-c%29%5E2%2B%28c-a%29%5E2%5D)
Hence proved.