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/sci/ - Science & Math

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10523587 No.10523587 [Reply] [Original]

[math]
\text{Let }H\text{ be the unit hemisphere } \{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}\text{, }C\text{ the}
\\
\text{unit circle }\{(x,y,0):x^2+y^2=1\}\text{, and }P\text{ the regular pentagon inscribed in}
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C\text{. Determine the surface area of that portion of }H\text{ lying over the planar}
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\text{region inside }P\text{, and write your answer in the form }A \sin\alpha + B \cos\beta\text{, where}
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A,B,\alpha,\beta\text{ are real numbers.}
[/math]

>> No.10523589

Previous Thread >>10520885

>> No.10523842

>>10523587
thinly veiled homework thread

>> No.10523918

>easier just to assume H is a full sphere and then divide the answer by 2
>suffices to find the area for one of the pentagon's five constituent triangles
>the "cut-off" is a plane (the vertical plane containing the triangle's outward-facing side), so its intersection with a sphere is a circle
>cut-off portion of H is subtended by angle (360/5)= 72 degrees, or really 36 degrees starting from its center
>cut-off area (still assuming H is a sphere) is therefore [math]int_0^{\pi/5} 2\pi \sin\theta d\theta = 2\pi - 2\pi\cos(\pi/5)[/math]
>multiply by 5/2
>get 0/10 score because they asked for [math]\sin\alpha[/math] in the answer

>> No.10523922
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10523922

>>10523918
>didn't take into account the slantiness of the sphere's surface when defining the integral
Oh shit

>> No.10524200

bump

>> No.10524229

>>10523842
based

>> No.10524232

bro please where ate the prismrivers

>> No.10524474

>>10523587
Imagine the spherical surface with r = 1, so that A = 4pi

The inscribed pentagon is like slicing off parts of the surface, each spherical sector beyond the pentagon contributes an area of 2pi*h, where h is the largest distance from the pentagon edge to the unit circle. Simple trig to find h = 1 - cos(pi/5)

So cutting 5 sectors off a full sphere this way has an area of: 4pi - 5*2pi(1 - cos(pi/5))

Remove half the area because of the hemisphere: 2pi - 5pi(1 - cos(pi/5))

Rearrange:
-3pi + 5pi cos(pi/5)

sin(pi/2) = 1 to fit the format of the answer so:
A = -3pi sin(pi/2) + 5pi cos(pi/5)

>> No.10525635

bump

>> No.10527697

>>10525635
Isn't>>10524474 the answer?