Please answer as quickly as you can! ♥ Find two possible dimensions for a cylinder that has a volume of 64π cm³.

Mathematics

1 answer:

11Alexandr11 [23.1K]3 years ago

3 0

Answer:

Possible dimensions are $r=4\,\,cm\,,\,h=4\,\,cm$ and $h=16\,\,cm\,,\,r=2\,\,cm$

Step-by-step explanation:

Given:

Volume of a cylinder is $64\pi\,\,cm^3$

To find: Dimensions of a cylinder

Solution:

Let $r,h$ denote height of a cylinder.

Volume of a cylinder = $\pi r^2h$

Therefore,

$64\pi=\pi r^2h\\64=r^2h\\4^2\,4=r^2h$

One possible dimension can be $r=4\,\,cm\,,\,h=4\,\,cm$ .

Also, $64=r^2h$ can be written as $16(2^2)=hr^2$

So, another possible dimension can be $h=16\,\,cm\,,\,r=2\,\,cm$

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Part 2: Use the information provided to write the vertex form equation of each parabola.

Serggg [28]

Answer: 1. x = -2(y - 4)² + 1

2. x = -y² + 5

3. y = -5(x + 1)² + 2

<u>Step-by-step explanation:</u>

Notes: The vertex formula of a parabola is x = a(y - k)² + h or y = a(x - h)² + k

(h, k) is the vertex
p is the distance from the vertex to the focus

$\bullet\quad a=\dfrac{1}{4p}$

$\text{Vertex}=(1,4)\qquad \text{Focus}:\bigg(\dfrac{7}{8},4\bigg)\\\\\text{Given}: (h,k)=(1,4)\\\\\\p=focus-vertex=\dfrac{7}{8}-\dfrac{8}{8}=\dfrac{-1}{8}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-1}{8})}=\dfrac{1}{-\frac{1}{2}}=-2$

Now input a = -2 and (h, k) = (1, 4) into the equation x = a(y - k)² + h

x = -2(y - 4)² + 1

***********************************************************************************

$\text{Vertex}=(5,0)\qquad \text{Directrix}:x=\dfrac{21}{4}\\\\\text{Given}: (h,k)=(5,0)\\\\\\p=vertex-directrix=\dfrac{20}{4}-\dfrac{21}{4}=\dfrac{-1}{4}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-1}{4})}=\dfrac{1}{-1}=-1$

Now input a = -1 and (h, k) = (5, 0) into the equation x = a(y - k)² + h

x = -1(y - 0)² + 5 → x = -y² + 5

***********************************************************************************

$\text{Vertex}=(-1,2)\qquad \text{Directrix}:y=\dfrac{41}{20}\\\\\text{Given}: (h,k)=(-1,2)\\\\\\p=vertex-directrix=\dfrac{40}{20}-\dfrac{41}{20}=\dfrac{-1}{20}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-1}{20})}=\dfrac{1}{-\frac{1}{5}}=-5$