Answer:
Step-by-step explanation:
Given
z=119+120 i
Let ![\sqrt{119+120 i}=p+iq](https://tex.z-dn.net/?f=%5Csqrt%7B119%2B120%20i%7D%3Dp%2Biq)
Squaring both sides
![119+120 i=p^2-q^2+2ipq](https://tex.z-dn.net/?f=119%2B120%20i%3Dp%5E2-q%5E2%2B2ipq)
Comparing real and imaginary part
Re(LHS)=Re(RHS)
-----------1
comparing Im(LHS)=Im(RHS)
120=2pq
![q=\frac{60}{p}](https://tex.z-dn.net/?f=q%3D%5Cfrac%7B60%7D%7Bp%7D)
Substitute q in 1
![119=p^2-(\frac{60}{p})^2](https://tex.z-dn.net/?f=119%3Dp%5E2-%28%5Cfrac%7B60%7D%7Bp%7D%29%5E2)
![p^4-119p^2-(68)^2=0](https://tex.z-dn.net/?f=p%5E4-119p%5E2-%2868%29%5E2%3D0)
Let ![x=p^2](https://tex.z-dn.net/?f=x%3Dp%5E2)
![x^2-119x-4624=0](https://tex.z-dn.net/?f=x%5E2-119x-4624%3D0)
![x=frac{119\pm \sqrt{119^2+4\times 4624}}{2}](https://tex.z-dn.net/?f=x%3Dfrac%7B119%5Cpm%20%5Csqrt%7B119%5E2%2B4%5Ctimes%204624%7D%7D%7B2%7D)
![x=\frac{119\pm 180.71}{2}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B119%5Cpm%20180.71%7D%7B2%7D)
we take only Positive value because ![p^2=x](https://tex.z-dn.net/?f=p%5E2%3Dx)
x=149.85
![p^2=149.85](https://tex.z-dn.net/?f=p%5E2%3D149.85)
thus ![p=\pm 12.24](https://tex.z-dn.net/?f=p%3D%5Cpm%2012.24)
![q=\mp 4.90](https://tex.z-dn.net/?f=q%3D%5Cmp%204.90)
thus ![\sqrt{119+120 i}=\pm (12.24+i 4.90)](https://tex.z-dn.net/?f=%5Csqrt%7B119%2B120%20i%7D%3D%5Cpm%20%2812.24%2Bi%204.90%29)
![y'+xy=xy^2\implies y^{-2}y'+xy^{-1}=x](https://tex.z-dn.net/?f=y%27%2Bxy%3Dxy%5E2%5Cimplies%20y%5E%7B-2%7Dy%27%2Bxy%5E%7B-1%7D%3Dx)
Let
![z=y^{-1}](https://tex.z-dn.net/?f=z%3Dy%5E%7B-1%7D)
, so that
![z'=-y^{-2}y'](https://tex.z-dn.net/?f=z%27%3D-y%5E%7B-2%7Dy%27)
. Then the ODE becomes linear in
![z](https://tex.z-dn.net/?f=z)
with
![-z'+xz=x\implies z'-xz=-x](https://tex.z-dn.net/?f=-z%27%2Bxz%3Dx%5Cimplies%20z%27-xz%3D-x)
Find an integrating factor:
![\mu(x)=\exp\left(\displaystyle\int-x\,\mathrm dx\right)=e^{-x^2/2}](https://tex.z-dn.net/?f=%5Cmu%28x%29%3D%5Cexp%5Cleft%28%5Cdisplaystyle%5Cint-x%5C%2C%5Cmathrm%20dx%5Cright%29%3De%5E%7B-x%5E2%2F2%7D)
Multiply both sides of the ODE by
![\mu](https://tex.z-dn.net/?f=%5Cmu)
:
![e^{-x^2/2}z'-xe^{-x^2/2}z=-xe^{-x^2/2}](https://tex.z-dn.net/?f=e%5E%7B-x%5E2%2F2%7Dz%27-xe%5E%7B-x%5E2%2F2%7Dz%3D-xe%5E%7B-x%5E2%2F2%7D)
The left side can be consolidated as a derivative:
![\left(e^{-x^2/2}z\right)'=-xe^{-x^2/2}](https://tex.z-dn.net/?f=%5Cleft%28e%5E%7B-x%5E2%2F2%7Dz%5Cright%29%27%3D-xe%5E%7B-x%5E2%2F2%7D)
Integrate both sides with respect to
![x](https://tex.z-dn.net/?f=x)
to get
![e^{-x^2/2}z=e^{x^2/2}+C](https://tex.z-dn.net/?f=e%5E%7B-x%5E2%2F2%7Dz%3De%5E%7Bx%5E2%2F2%7D%2BC)
where the right side can be computed with a simple substitution. Then
![z=1+Ce^{x^2/2}](https://tex.z-dn.net/?f=z%3D1%2BCe%5E%7Bx%5E2%2F2%7D)
Back-substitute to solve for
![y](https://tex.z-dn.net/?f=y)
.
Answer:
Second answer
Step-by-step explanation:
ANSWER: 312 3/8
So Volume is Length • Width • Height
So first we would put it into this formula:
8.5 • 5.25 • 7= 312.375, or, 312 3/8
Answer:
y = 4/3x - 10
Step-by-step explanation:
Slope: 4/3
y-intercept: -2 - (4/3)(6) = -2 - 8 = -10