I wanna say 14%, just because if you add the total of pieces divided by the 5 yellow with 4 holes, you'll receive the answer 0.14.
Answer:
![1 \rightarrow E, 2\rightarrow B, 3\rightarrow C, 4\rightarrow D, 5\rightarrow A](https://tex.z-dn.net/?f=1%20%5Crightarrow%20E%2C%202%5Crightarrow%20B%2C%203%5Crightarrow%20C%2C%204%5Crightarrow%20D%2C%205%5Crightarrow%20A)
Step-by-step explanation:
1. ![\frac{d^2y}{dx^2}+25y=0](https://tex.z-dn.net/?f=%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%2B25y%3D0)
The characteristic equation for the given differential equation is:
![r^{2} +25=0](https://tex.z-dn.net/?f=r%5E%7B2%7D%20%2B25%3D0)
![\Rightarrow r^2=-25](https://tex.z-dn.net/?f=%5CRightarrow%20r%5E2%3D-25)
![\Rightarrow r=\pm 5i](https://tex.z-dn.net/?f=%5CRightarrow%20r%3D%5Cpm%20%205i)
Since the roots are complex
Now, the general solution is:
![y=A\cos 5x+B\sin 5x](https://tex.z-dn.net/?f=y%3DA%5Ccos%205x%2BB%5Csin%205x)
2. ![\frac{dy}{dx}=\frac {2xy}{x^2}-5y^2](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%20%7B2xy%7D%7Bx%5E2%7D-5y%5E2)
![\Rightarrow \frac{dy}{dx}-\frac 2xy=-5y^2](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cfrac%7Bdy%7D%7Bdx%7D-%5Cfrac%202xy%3D-5y%5E2)
Divide both sides by ![y^{-1}](https://tex.z-dn.net/?f=y%5E%7B-1%7D)
Let, ![v=y^{-1} \Rightarrow \frac{dv}{dx}=-y^{-2}\frac{dy}{dx}](https://tex.z-dn.net/?f=v%3Dy%5E%7B-1%7D%20%5CRightarrow%20%5Cfrac%7Bdv%7D%7Bdx%7D%3D-y%5E%7B-2%7D%5Cfrac%7Bdy%7D%7Bdx%7D)
![\Rightarrow -\frac{dv}{dx}-\frac 2xv=-5](https://tex.z-dn.net/?f=%5CRightarrow%20-%5Cfrac%7Bdv%7D%7Bdx%7D-%5Cfrac%202xv%3D-5)
![\Rightarrow \frac{dv}{dx}+\frac 2xv=5](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cfrac%7Bdv%7D%7Bdx%7D%2B%5Cfrac%202xv%3D5)
Here, ![p(x)=\frac 2x\; \text{and}\;\; q(x)=5](https://tex.z-dn.net/?f=p%28x%29%3D%5Cfrac%202x%5C%3B%20%5Ctext%7Band%7D%5C%3B%5C%3B%20q%28x%29%3D5)
I.F. ![=e^{\int \frac 2xdx}=x^2](https://tex.z-dn.net/?f=%3De%5E%7B%5Cint%20%5Cfrac%202xdx%7D%3Dx%5E2)
Now, the general solution is:
![vx^2=\int x^2 5dx=\frac {5x^3}3+c](https://tex.z-dn.net/?f=vx%5E2%3D%5Cint%20x%5E2%205dx%3D%5Cfrac%20%7B5x%5E3%7D3%2Bc)
![\Rightarrow \frac {x^2}y-\frac {5x^3}3=c](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cfrac%20%7Bx%5E2%7Dy-%5Cfrac%20%7B5x%5E3%7D3%3Dc)
![\Rightarrow 3x^2-5x^3y=Cy](https://tex.z-dn.net/?f=%5CRightarrow%203x%5E2-5x%5E3y%3DCy)
3. ![\frac{d^2y}{dx^2}+16\frac{dy}{dx}+64y=0](https://tex.z-dn.net/?f=%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%2B16%5Cfrac%7Bdy%7D%7Bdx%7D%2B64y%3D0)
The characteristic equation is:
![r^2+16r+64=0](https://tex.z-dn.net/?f=r%5E2%2B16r%2B64%3D0)
![\Rightarrow r^2+8r+8r+64=0](https://tex.z-dn.net/?f=%5CRightarrow%20r%5E2%2B8r%2B8r%2B64%3D0)
![\Rightarrow r(r+8)+8(r+8)=0](https://tex.z-dn.net/?f=%5CRightarrow%20r%28r%2B8%29%2B8%28r%2B8%29%3D0)
![\Rightarrow (r+8)(r+8)=0](https://tex.z-dn.net/?f=%5CRightarrow%20%28r%2B8%29%28r%2B8%29%3D0)
![\Rightarrow r=-8,-8](https://tex.z-dn.net/?f=%5CRightarrow%20r%3D-8%2C-8)
Since the roots are real and repeated.
Now, the general solution is:
![y=Ae^{-8x}+Bxe^{-8x}](https://tex.z-dn.net/?f=y%3DAe%5E%7B-8x%7D%2BBxe%5E%7B-8x%7D)
4. ![\frac {dy}{dx}=10xy](https://tex.z-dn.net/?f=%5Cfrac%20%7Bdy%7D%7Bdx%7D%3D10xy)
![\Rightarrow \frac {dy}{y}=10xdx](https://tex.z-dn.net/?f=%5CRightarrow%20%20%5Cfrac%20%7Bdy%7D%7By%7D%3D10xdx)
Integrating both sides
![\int\frac {dy}y=\int 10xdx+\log c](https://tex.z-dn.net/?f=%5Cint%5Cfrac%20%7Bdy%7Dy%3D%5Cint%2010xdx%2B%5Clog%20c)
![\Rightarrow \log y=5x^2+\log c](https://tex.z-dn.net/?f=%5CRightarrow%20%5Clog%20y%3D5x%5E2%2B%5Clog%20c)
5. ![\frac {dy}{dx}+24x^2y=24x^2](https://tex.z-dn.net/?f=%5Cfrac%20%7Bdy%7D%7Bdx%7D%2B24x%5E2y%3D24x%5E2)
Here, ![p(x)=24x^2 \; \text{and}\;\; q(x)=24x^2](https://tex.z-dn.net/?f=p%28x%29%3D24x%5E2%20%5C%3B%20%5Ctext%7Band%7D%5C%3B%5C%3B%20q%28x%29%3D24x%5E2)
I.F.![= e^{\int 24x^2dx}=e^{8x^3}](https://tex.z-dn.net/?f=%3D%20e%5E%7B%5Cint%2024x%5E2dx%7D%3De%5E%7B8x%5E3%7D)
Now, the general solution is:
![y.e^{8x^3}=\int 24x^2 e^{8x^3}dx=24\int x^2e^{8x^3}dx](https://tex.z-dn.net/?f=y.e%5E%7B8x%5E3%7D%3D%5Cint%2024x%5E2%20e%5E%7B8x%5E3%7Ddx%3D24%5Cint%20x%5E2e%5E%7B8x%5E3%7Ddx)
Let, ![8x^3=t \Rightarrow 24x^2dx=dt\Rightarrow x^2dx=\frac {dt}{24}](https://tex.z-dn.net/?f=8x%5E3%3Dt%20%5CRightarrow%2024x%5E2dx%3Ddt%5CRightarrow%20x%5E2dx%3D%5Cfrac%20%7Bdt%7D%7B24%7D)
![\Rightarrow ye^{8x^3}=\int e^tdt](https://tex.z-dn.net/?f=%5CRightarrow%20ye%5E%7B8x%5E3%7D%3D%5Cint%20e%5Etdt)
![\Rightarrow ye^{8x^3}=e^{8x^3}+c](https://tex.z-dn.net/?f=%5CRightarrow%20ye%5E%7B8x%5E3%7D%3De%5E%7B8x%5E3%7D%2Bc)
![\Rightarrow y=1+ce^{-8x^3}](https://tex.z-dn.net/?f=%5CRightarrow%20y%3D1%2Bce%5E%7B-8x%5E3%7D)
Answer:
Step-by-step explanation:
5⁸ ÷ 5⁶ = 5⁽⁸⁻⁶⁾ = 5² = 5*5 = 25
or
Well, that means there are 3*9=27 cals from fat. 27/150=18% of calories.