Two end points
One direction
Hope this helps xx
Answer:
x=15√2 y=90 degrees.
Step-by-step explanation:
Thanks to the Pythagorean Theorem we know that with any right triangle, A^2+B^2=C^2. In this case we can use 15^2+15^2=sqrt of 450. The square root of 450 is 15 times the square root of 2. Thus x=15√2. Now we need to find y. All angles in a triangle add up to 180 so if we have 45+45+y=180 then y=90.
The cone equation gives
![z=\sqrt{x^2+y^2}\implies z^2=x^2+y^2](https://tex.z-dn.net/?f=z%3D%5Csqrt%7Bx%5E2%2By%5E2%7D%5Cimplies%20z%5E2%3Dx%5E2%2By%5E2)
which means that the intersection of the cone and sphere occurs at
![x^2+y^2+(x^2+y^2)=9\implies x^2+y^2=\dfrac92](https://tex.z-dn.net/?f=x%5E2%2By%5E2%2B%28x%5E2%2By%5E2%29%3D9%5Cimplies%20x%5E2%2By%5E2%3D%5Cdfrac92)
i.e. along the vertical cylinder of radius
![\dfrac3{\sqrt2}](https://tex.z-dn.net/?f=%5Cdfrac3%7B%5Csqrt2%7D)
when
![z=\dfrac3{\sqrt2}](https://tex.z-dn.net/?f=z%3D%5Cdfrac3%7B%5Csqrt2%7D)
.
We can parameterize the spherical cap in spherical coordinates by
![\mathbf r(\theta,\varphi)=\langle3\cos\theta\sin\varphi,3\sin\theta\sin\varphi,3\cos\varphi\right\rangle](https://tex.z-dn.net/?f=%5Cmathbf%20r%28%5Ctheta%2C%5Cvarphi%29%3D%5Clangle3%5Ccos%5Ctheta%5Csin%5Cvarphi%2C3%5Csin%5Ctheta%5Csin%5Cvarphi%2C3%5Ccos%5Cvarphi%5Cright%5Crangle)
where
![0\le\theta\le2\pi](https://tex.z-dn.net/?f=0%5Cle%5Ctheta%5Cle2%5Cpi)
and
![0\le\varphi\le\dfrac\pi4](https://tex.z-dn.net/?f=0%5Cle%5Cvarphi%5Cle%5Cdfrac%5Cpi4)
, which follows from the fact that the radius of the sphere is 3 and the height at which the sphere and cone intersect is
![\dfrac3{\sqrt2}](https://tex.z-dn.net/?f=%5Cdfrac3%7B%5Csqrt2%7D)
. So the angle between the vertical line through the origin and any line through the origin normal to the sphere along the cone's surface is
![\varphi=\cos^{-1}\left(\dfrac{\frac3{\sqrt2}}3\right)=\cos^{-1}\left(\dfrac1{\sqrt2}\right)=\dfrac\pi4](https://tex.z-dn.net/?f=%5Cvarphi%3D%5Ccos%5E%7B-1%7D%5Cleft%28%5Cdfrac%7B%5Cfrac3%7B%5Csqrt2%7D%7D3%5Cright%29%3D%5Ccos%5E%7B-1%7D%5Cleft%28%5Cdfrac1%7B%5Csqrt2%7D%5Cright%29%3D%5Cdfrac%5Cpi4)
Now the surface area of the cap is given by the surface integral,
![\displaystyle\iint_{\text{cap}}\mathrm dS=\int_{\theta=0}^{\theta=2\pi}\int_{\varphi=0}^{\varphi=\pi/4}\|\mathbf r_u\times\mathbf r_v\|\,\mathrm dv\,\mathrm du](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_%7B%5Ctext%7Bcap%7D%7D%5Cmathrm%20dS%3D%5Cint_%7B%5Ctheta%3D0%7D%5E%7B%5Ctheta%3D2%5Cpi%7D%5Cint_%7B%5Cvarphi%3D0%7D%5E%7B%5Cvarphi%3D%5Cpi%2F4%7D%5C%7C%5Cmathbf%20r_u%5Ctimes%5Cmathbf%20r_v%5C%7C%5C%2C%5Cmathrm%20dv%5C%2C%5Cmathrm%20du)
![=\displaystyle\int_{u=0}^{u=2\pi}\int_{\varphi=0}^{\varphi=\pi/4}9\sin v\,\mathrm dv\,\mathrm du](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cint_%7Bu%3D0%7D%5E%7Bu%3D2%5Cpi%7D%5Cint_%7B%5Cvarphi%3D0%7D%5E%7B%5Cvarphi%3D%5Cpi%2F4%7D9%5Csin%20v%5C%2C%5Cmathrm%20dv%5C%2C%5Cmathrm%20du)
![=-18\pi\cos v\bigg|_{v=0}^{v=\pi/4}](https://tex.z-dn.net/?f=%3D-18%5Cpi%5Ccos%20v%5Cbigg%7C_%7Bv%3D0%7D%5E%7Bv%3D%5Cpi%2F4%7D)
![=18\pi\left(1-\dfrac1{\sqrt2}\right)](https://tex.z-dn.net/?f=%3D18%5Cpi%5Cleft%281-%5Cdfrac1%7B%5Csqrt2%7D%5Cright%29)
Answer:
each time the numbers are being multiplied by 10
Step-by-step explanation:
1 × 10 = 10
10 × 10 = 100
100 × 10 = 1,000
hope this helps :)