She is 3,457 kilometers from her old house. 1 kilometer=1000 meters
As the hint suggests, convert to spherical coordinates using
x = p cos(u) sin(v)
y = p sin(u) sin(v)
z = p cos(v)
dV = dx dy dz = p² sin(v) dp du dv
Then U is the set
![U = \left\{ (p,u,v) \mid 0\le p\le2 \text{ and } 0\le u\le \dfrac{\pi}2 \text{ and } 0\le v\le\dfrac{\pi}2\right\}](https://tex.z-dn.net/?f=U%20%3D%20%5Cleft%5C%7B%20%28p%2Cu%2Cv%29%20%5Cmid%200%5Cle%20p%5Cle2%20%5Ctext%7B%20and%20%7D%200%5Cle%20u%5Cle%20%5Cdfrac%7B%5Cpi%7D2%20%5Ctext%7B%20and%20%7D%200%5Cle%20v%5Cle%5Cdfrac%7B%5Cpi%7D2%5Cright%5C%7D)
and the integral of y over U is
![\displaystyle \iiint_U y \, dV = \iiint_U p\sin(u)\sin(v) \cdot p^2 \sin(v) \, dV](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ciiint_U%20y%20%5C%2C%20dV%20%3D%20%5Ciiint_U%20p%5Csin%28u%29%5Csin%28v%29%20%5Ccdot%20p%5E2%20%5Csin%28v%29%20%5C%2C%20dV)
![\displaystyle \iiint_U y \, dV = \int_0^{\frac\pi2} \int_0^{\frac\pi2} \int_0^2 p^3 \sin(u) \sin^2(v) \, dp \, du \, dv](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ciiint_U%20y%20%5C%2C%20dV%20%3D%20%20%5Cint_0%5E%7B%5Cfrac%5Cpi2%7D%20%5Cint_0%5E%7B%5Cfrac%5Cpi2%7D%20%5Cint_0%5E2%20p%5E3%20%5Csin%28u%29%20%5Csin%5E2%28v%29%20%5C%2C%20dp%20%5C%2C%20du%20%5C%2C%20dv%20)
![\displaystyle \iiint_U y \, dV = \frac{2^4-0^4}4 \int_0^{\frac\pi2} \int_0^{\frac\pi2} \sin(u) \sin^2(v) \, du \, dv](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%5Ciiint_U%20y%20%5C%2C%20dV%20%3D%20%5Cfrac%7B2%5E4-0%5E4%7D4%20%5Cint_0%5E%7B%5Cfrac%5Cpi2%7D%20%5Cint_0%5E%7B%5Cfrac%5Cpi2%7D%20%5Csin%28u%29%20%5Csin%5E2%28v%29%20%5C%2C%20du%20%5C%2C%20dv%20)
![\displaystyle \iiint_U y \, dV = 4 \cdot \left(-\cos\left(\frac\pi2\right) + \cos(0)\right) \int_0^{\frac\pi2} \sin^2(v) \, dv](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ciiint_U%20y%20%5C%2C%20dV%20%3D%20%204%20%5Ccdot%20%5Cleft%28-%5Ccos%5Cleft%28%5Cfrac%5Cpi2%5Cright%29%20%2B%20%5Ccos%280%29%5Cright%29%20%5Cint_0%5E%7B%5Cfrac%5Cpi2%7D%20%5Csin%5E2%28v%29%20%5C%2C%20dv%20)
![\displaystyle \iiint_U y \, dV = 4 \cdot \frac12 \int_0^{\frac\pi2} (1-\cos(2v)) \, dv](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ciiint_U%20y%20%5C%2C%20dV%20%3D%20%204%20%5Ccdot%20%5Cfrac12%20%5Cint_0%5E%7B%5Cfrac%5Cpi2%7D%20%281-%5Ccos%282v%29%29%20%5C%2C%20dv%20)
![\displaystyle \iiint_U y \, dV = 2 \left(\left(\frac\pi2 - \frac12 \sin\left(2\cdot\frac\pi2\right)\right) - \left(0 - \frac12 \sin\left(2\cdot0\right)\right) \right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ciiint_U%20y%20%5C%2C%20dV%20%3D%20%202%20%5Cleft%28%5Cleft%28%5Cfrac%5Cpi2%20-%20%5Cfrac12%20%5Csin%5Cleft%282%5Ccdot%5Cfrac%5Cpi2%5Cright%29%5Cright%29%20-%20%5Cleft%280%20-%20%5Cfrac12%20%5Csin%5Cleft%282%5Ccdot0%5Cright%29%5Cright%29%20%5Cright%29)
![\displaystyle \iiint_U y \, dV = \pi - \sin(\pi) = \boxed{\pi}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ciiint_U%20y%20%5C%2C%20dV%20%3D%20%20%5Cpi%20-%20%5Csin%28%5Cpi%29%20%3D%20%5Cboxed%7B%5Cpi%7D)
The answer is 3x^2-12=-12, imiplies <span>3x^2 = - 12 + 12 = 0
so </span><span>3x^2 = 0 implies x = 0
there is one solution, x =0</span>
Use symmetry and find z so that probability is:
(100-39)/2
=61%
=30.5%~0.305
From the z-table:
z=-0.51
Answer: -0.51
Answer:
y=63, x=20, x=47, b=70, c=89, x=133
Step-by-step explanation:
1) y=63- alternate angles are the same
2) 4x+5x=180
9x=180
x=20
3) x=47
4) b=70- angles on a line equal to 180
c=89- angles in a triangle add up to 180
5) x=133- angles that are alternate are the same