Mathematics, 25.09.2019 13:20 catwomn

Find the area of the helicoid (or spiral ramp) with vector equation r(u, v) = ucos(v) i + usin(v) j + v k, 0 ≤ u ≤ 1, 0 ≤ v ≤ 9π.

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Answer from: tyliyahmiles99

The area f the helicoid ramp is:

∫∫A) | r(u) *r( v) | dudv

The solution is:

A = 10.35×π square units

r ( u , v ) = u×cosv i + u×sinv j +v k

To get

r (u ) = δ(r ( u , v ) ) / δu = [ cosv , sinv , 0 ]

r ( v ) = δ(r ( u , v ) ) / δv = [ -u×sinv , u×cosv , 1 ]

The vectorial product is:

i j k

r (u ) * r ( v ) cosv sinv 0

-u×sinv u×cosv 1

r (u ) * r ( v ) = i × ( sinv - 0 ) - j × ( cosv - 0 ) + k ( u×cos²v + u× sin²v )

r (u ) * r ( v ) = sinv i - cosv j + u k

Now

| r (u ) * r ( v ) | = √sin²v + cos²v + u² = √ 1 + u²

Then

A = ∫₀ (9π) dv ∫₀¹ √ 1 + u² du

∫₀¹ √ 1 + u² du = 1.15

A = 1.15 × v |( 0 , 9π )

A = 10.35×π square units

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Answer from: 191600

Let

denote the helicoid parameterized by

$\mathbb r(u,v)=u\cos v\,\mathbf i+u\sin v\,\mathbf j+v\,\mathbf k$

for $0\le u\le1$ and $0\le v\le9\pi$ . The surface area is given by the surface integral,

$\displaystyle\iint_H\mathrm dS=\iint_H\|\mathbf r_u\times\mathbf r_v\|\,\mathrm du\,\mathrm dv$

We have

$\mathbf r_u=\dfrac{\partial\mathbf r(u,v)}{\partial u}=\cos v\,\mathbf i+\sin v\,\mathbf j$
$\mathbf r_v=\dfrac{\partial\mathbf r(u,v)}{\partial v}=-u\sin v\,\mathbf i+u\cos v\,\mathbf j+\mathbf k$
$\implies\mathbf r_u\times\mathbf r_v=\sin v\,\mathbf i-\cos v\,\mathbf j+u\,\mathbf k$
$\implies\|\mathbf r_u\times\mathbf r_v\|=\sqrt{1+u^2}$

So the area of

is

$\displaystyle\iint_H\mathrm dS=\int_{v=0}^{v=9\pi}\int_{u=0}^{u=1}\sqrt{1+u^2}\,\mathrm du\,\mathrm dv$
$=\dfrac{9(\sqrt2+\sinh^{-1}(1))\pi}2$

Answer from: Quest

what are ur choices?

Answer from: Quest

median is the value separating the higher half of a data sample, a population or probablity distribution from the lower half.

step-by-step explanation:

say you have 3,8,7,3,1,9,8 order them from lowest to highest

if orderd it would be 1,3,3,7,8,8,9 now you take away from the lowest and the highest until you end up in the middle.

so you would remove the 1 and 9 leaving you with 5 digits remaining

when you've reached the middle that would be your answer so in this case the answer is 7

if there were two numbers left say 7 and 5 you'd and them together so 12 then divide by 2 witch would equal 6 so the answer would be 6

good luck on that project hoped i you at all

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Find the area of the helicoid (or spiral ramp) with vector equation r(u, v) = ucos(v) i + usin(v) j...

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