Answer:
h = 6 cm
Step-by-step explanation:
Volume of cylinder = 54 π cm³
πr²h = 54π
π * 3 * 3 * h = 54π
![h = \frac{54 \pi }{\pi *3*3}\\\\h= 6](https://tex.z-dn.net/?f=h%20%3D%20%5Cfrac%7B54%20%5Cpi%20%7D%7B%5Cpi%20%2A3%2A3%7D%5C%5C%5C%5Ch%3D%206)
Please restore my faith in humans and try to understand this concept
Answer: 3
Step-by-step explanation:
7-3+4-2+9=15
15/5=3
Answer:
Step-by-step explanation:
![\frac{dy}{dx}=x^{2}(y-1)\\\frac{1}{y-1} \text{ } dy=x^{2} \text{ } dx\\\int \frac{1}{y-1} \text{ } dy=\int x^{2} \text{ } dx\\\ln|y-1|=\frac{x^{3}}{3}+C\\](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%3Dx%5E%7B2%7D%28y-1%29%5C%5C%5Cfrac%7B1%7D%7By-1%7D%20%5Ctext%7B%20%7D%20dy%3Dx%5E%7B2%7D%20%5Ctext%7B%20%7D%20dx%5C%5C%5Cint%20%5Cfrac%7B1%7D%7By-1%7D%20%5Ctext%7B%20%7D%20dy%3D%5Cint%20x%5E%7B2%7D%20%5Ctext%7B%20%7D%20dx%5C%5C%5Cln%7Cy-1%7C%3D%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D%2BC%5C%5C)
From the initial condition,
![\ln|3-1|=\frac{0^{3}}{3}+C\\\ln 2=C](https://tex.z-dn.net/?f=%5Cln%7C3-1%7C%3D%5Cfrac%7B0%5E%7B3%7D%7D%7B3%7D%2BC%5C%5C%5Cln%202%3DC)
So we have that ![\ln |y-1|=\frac{x^{3}}{3}+\ln 2\\e^{\frac{x^{3}}{3}+\ln 2}=y-1\\2e^{\frac{x^{3}}{3}}=y-1\\y=2e^{\frac{x^{3}}{3}}+1](https://tex.z-dn.net/?f=%5Cln%20%7Cy-1%7C%3D%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D%2B%5Cln%202%5C%5Ce%5E%7B%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D%2B%5Cln%202%7D%3Dy-1%5C%5C2e%5E%7B%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D%7D%3Dy-1%5C%5Cy%3D2e%5E%7B%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D%7D%2B1)
Answer:
case a)
----> open up
case b)
----> open down
case c)
----> open left
case d)
----> open right
Step-by-step explanation:
we know that
1) The general equation of a vertical parabola is equal to
![y=a(x-h)^{2}+k](https://tex.z-dn.net/?f=y%3Da%28x-h%29%5E%7B2%7D%2Bk)
where
a is a coefficient
(h,k) is the vertex
If a>0 ----> the parabola open upward and the vertex is a minimum
If a<0 ----> the parabola open downward and the vertex is a maximum
2) The general equation of a horizontal parabola is equal to
![x=a(y-k)^{2}+h](https://tex.z-dn.net/?f=x%3Da%28y-k%29%5E%7B2%7D%2Bh)
where
a is a coefficient
(h,k) is the vertex
If a>0 ----> the parabola open to the right
If a<0 ----> the parabola open to the left
Verify each case
case a) we have
![x^{2}=3y](https://tex.z-dn.net/?f=x%5E%7B2%7D%3D3y)
so
![y=(1/3)x^{2}](https://tex.z-dn.net/?f=y%3D%281%2F3%29x%5E%7B2%7D)
![a=(1/3)](https://tex.z-dn.net/?f=a%3D%281%2F3%29)
so
![a>0](https://tex.z-dn.net/?f=a%3E0)
therefore
The parabola open up
case b) we have
![x^{2}=-10y](https://tex.z-dn.net/?f=x%5E%7B2%7D%3D-10y)
so
![y=-(1/10)x^{2}](https://tex.z-dn.net/?f=y%3D-%281%2F10%29x%5E%7B2%7D)
![a=-(1/10)](https://tex.z-dn.net/?f=a%3D-%281%2F10%29)
![a](https://tex.z-dn.net/?f=a%3C0)
therefore
The parabola open down
case c) we have
![y^{2}=-2x](https://tex.z-dn.net/?f=y%5E%7B2%7D%3D-2x)
so
![x=-(1/2)y^{2}](https://tex.z-dn.net/?f=x%3D-%281%2F2%29y%5E%7B2%7D)
![a=-(1/2)](https://tex.z-dn.net/?f=a%3D-%281%2F2%29)
![a](https://tex.z-dn.net/?f=a%3C0)
therefore
The parabola open to the left
case d) we have
![y^{2}=6x](https://tex.z-dn.net/?f=y%5E%7B2%7D%3D6x)
so
![x=(1/6)y^{2}](https://tex.z-dn.net/?f=x%3D%281%2F6%29y%5E%7B2%7D)
![a=(1/6)](https://tex.z-dn.net/?f=a%3D%281%2F6%29)
![a>0](https://tex.z-dn.net/?f=a%3E0)
therefore
The parabola open to the right