Answer:
The answer is ![\sqrt{\frac{6}{5}}](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7B6%7D%7B5%7D%7D)
Step-by-step explanation:
To calculate the volumen of the solid we solve the next double integral:
![\int\limits^1_0\int\limits^1_0 {12xy^{2} } \, dxdy](https://tex.z-dn.net/?f=%5Cint%5Climits%5E1_0%5Cint%5Climits%5E1_0%20%7B12xy%5E%7B2%7D%20%7D%20%5C%2C%20dxdy)
Solving:
![\int\limits^1_0 {12x} \, dx \int\limits^1_0 {y^{2} } \, dy](https://tex.z-dn.net/?f=%5Cint%5Climits%5E1_0%20%7B12x%7D%20%5C%2C%20dx%20%5Cint%5Climits%5E1_0%20%7By%5E%7B2%7D%20%7D%20%5C%2C%20dy)
![[6x^{2} ]{{1} \atop {0}} \right. * [\frac{y^{3}}{3}]{{1} \atop {0}} \right.](https://tex.z-dn.net/?f=%5B6x%5E%7B2%7D%20%5D%7B%7B1%7D%20%5Catop%20%7B0%7D%7D%20%5Cright.%20%2A%20%5B%5Cfrac%7By%5E%7B3%7D%7D%7B3%7D%5D%7B%7B1%7D%20%5Catop%20%7B0%7D%7D%20%5Cright.)
Replacing the limits:
![6*\frac{1}{3} =2](https://tex.z-dn.net/?f=6%2A%5Cfrac%7B1%7D%7B3%7D%20%3D2)
The plane y=mx divides this volume in two equal parts. So volume of one part is 1.
Since m > 1, hence mx ≤ y ≤ 1, 0 ≤ x ≤ ![\frac{1}{m}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bm%7D)
Solving the double integral with these new limits we have:
![\int\limits^\frac{1}{m} _0\int\limits^{1}_{mx} {12xy^{2} } \, dxdy](https://tex.z-dn.net/?f=%5Cint%5Climits%5E%5Cfrac%7B1%7D%7Bm%7D%20_0%5Cint%5Climits%5E%7B1%7D_%7Bmx%7D%20%7B12xy%5E%7B2%7D%20%7D%20%5C%2C%20dxdy)
This part is a little bit tricky so let's solve the integral first for dy:
![\int\limits^\frac{1}{m}_0 [{12x \frac{y^{3}}{3}}]{{1} \atop {mx}} \right.\, dx =\int\limits^\frac{1}{m}_0 [{4x y^{3 }]{{1} \atop {mx}} \right.\, dx](https://tex.z-dn.net/?f=%5Cint%5Climits%5E%5Cfrac%7B1%7D%7Bm%7D_0%20%5B%7B12x%20%5Cfrac%7By%5E%7B3%7D%7D%7B3%7D%7D%5D%7B%7B1%7D%20%5Catop%20%7Bmx%7D%7D%20%5Cright.%5C%2C%20dx%20%3D%5Cint%5Climits%5E%5Cfrac%7B1%7D%7Bm%7D_0%20%5B%7B4x%20y%5E%7B3%20%7D%5D%7B%7B1%7D%20%5Catop%20%7Bmx%7D%7D%20%5Cright.%5C%2C%20dx)
Replacing the limits:
![\int\limits^\frac{1}{m}_0 {4x(1-(mx)^{3} )\, dx =\int\limits^\frac{1}{m}_0 {4x-4x(m^{3} x^{3} )\, dx =\int\limits^\frac{1}{m}_0 ({4x-4m^{3} x^{4}) \, dx](https://tex.z-dn.net/?f=%5Cint%5Climits%5E%5Cfrac%7B1%7D%7Bm%7D_0%20%7B4x%281-%28mx%29%5E%7B3%7D%20%29%5C%2C%20dx%20%3D%5Cint%5Climits%5E%5Cfrac%7B1%7D%7Bm%7D_0%20%7B4x-4x%28m%5E%7B3%7D%20x%5E%7B3%7D%20%29%5C%2C%20dx%20%3D%5Cint%5Climits%5E%5Cfrac%7B1%7D%7Bm%7D_0%20%28%7B4x-4m%5E%7B3%7D%20x%5E%7B4%7D%29%20%5C%2C%20dx)
Solving now for dx:
![[{\frac{4x^{2}}{2} -\frac{4m^{3} x^{5}}{5} ]{{\frac{1}{m} } \atop {0}} \right. = [{2x^{2} -\frac{4m^{3} x^{5}}{5} ]{{\frac{1}{m} } \atop {0}} \right.](https://tex.z-dn.net/?f=%5B%7B%5Cfrac%7B4x%5E%7B2%7D%7D%7B2%7D%20-%5Cfrac%7B4m%5E%7B3%7D%20x%5E%7B5%7D%7D%7B5%7D%20%5D%7B%7B%5Cfrac%7B1%7D%7Bm%7D%20%7D%20%5Catop%20%7B0%7D%7D%20%5Cright.%20%3D%20%5B%7B2x%5E%7B2%7D%20-%5Cfrac%7B4m%5E%7B3%7D%20x%5E%7B5%7D%7D%7B5%7D%20%5D%7B%7B%5Cfrac%7B1%7D%7Bm%7D%20%7D%20%5Catop%20%7B0%7D%7D%20%5Cright.)
Replacing the limits:
![\frac{2}{m^{2} }-\frac{4m^{3}\frac{1}{m^{5}}}{5} =\frac{2}{m^{2} }-\frac{4\frac{1}{m^{2}}}{5} \\ \frac{2}{m^{2} }-\frac{4}{5m^{2} }=\frac{10m^{2}-4m^{2} }{5m^{4}} \\ \frac{6m^{2} }{5m^{4}} =\frac{6}{5m^{2}}](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7Bm%5E%7B2%7D%20%7D-%5Cfrac%7B4m%5E%7B3%7D%5Cfrac%7B1%7D%7Bm%5E%7B5%7D%7D%7D%7B5%7D%20%3D%5Cfrac%7B2%7D%7Bm%5E%7B2%7D%20%7D-%5Cfrac%7B4%5Cfrac%7B1%7D%7Bm%5E%7B2%7D%7D%7D%7B5%7D%20%5C%5C%20%5Cfrac%7B2%7D%7Bm%5E%7B2%7D%20%7D-%5Cfrac%7B4%7D%7B5m%5E%7B2%7D%20%7D%3D%5Cfrac%7B10m%5E%7B2%7D-4m%5E%7B2%7D%20%7D%7B5m%5E%7B4%7D%7D%20%5C%5C%20%5Cfrac%7B6m%5E%7B2%7D%20%7D%7B5m%5E%7B4%7D%7D%20%3D%5Cfrac%7B6%7D%7B5m%5E%7B2%7D%7D)
As I mentioned before, this volume is equal to 1, hence:
![\frac{6}{5m^{2}}=1\\m^{2} =\frac{6}{5} \\m=\sqrt{\frac{6}{5} }](https://tex.z-dn.net/?f=%5Cfrac%7B6%7D%7B5m%5E%7B2%7D%7D%3D1%5C%5Cm%5E%7B2%7D%20%3D%5Cfrac%7B6%7D%7B5%7D%20%5C%5Cm%3D%5Csqrt%7B%5Cfrac%7B6%7D%7B5%7D%20%7D)
8 children because
11 3/7 is equal to 83/14
If each child receives 10/14
83/10
= 8.3
There can't be .3 of a child so round it and you get 8!
Hope this helps!
Answer: 125.5 is the answer in decimal form. If you need it in fraction form it would be 251/2
To get this answer, you add up the parallel sides AB+DC = 69+182 =251
Then you divide by 2 leading us to get 251/2 = 125.5
The midsegment is basically the average of the two parallel sides
Answer:
40/140 ≈ 0,2857 corresponds 28,57%.