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malfutka [58]
2 years ago
12

Can I see detailed calculation for Dash?

Mathematics
1 answer:
egoroff_w [7]2 years ago
8 0
I think you can, sorry if i’m wrong
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Step-by-step explanation:

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The formula to find the amount is

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Subtracting 1 from both sides

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on 20 of 22 Which equation describes the line with slope 8 that contains the point (-3, -6)? O A. y + 3 = 8(x+6) O B. y-3 = 8(x-
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Hi, I just wanted to say that I got this question a few days ago on my test!

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2 years ago
The hemisphere of radius r is made from a stack of very thin plates such that the density varies with height, r = kz, where k is
Papessa [141]

Answer:

M = ¼ k π R⁴

zG = 8/15 R

Step-by-step explanation:

Note: I'm using lower case r as the radius of each plate and upper case R as the radius of the hemisphere.

The mass of each plate is density times volume:

dm = ρ dV

Each plate has a radius r and a thickness dz.  So the volume of each plate is:

dV = π r² dz

Substituting:

dm = ρ π r² dz

We're told that ρ = kz.  Substituting:

dm = kz π r² dz

Next, we need to write the radius r in terms of the height z.  To do that, we need to look at the cross section (see image below).

The height z and the radius r form a right triangle, where the hypotenuse is the radius of the hemisphere R.

Using Pythagorean theorem:

z² + r² = R²

r² = R² − z²

Substituting:

dm = kπ z (R² − z²) dz

We now have the mass of each plate as a function of its height.  To find the total mass, we integrate between z=0 and z=R.

M = ∫ dm

M = ∫₀ᴿ  kπ z (R² − z²) dz

M = kπ ∫₀ᴿ (R² z − z³) dz

M = kπ (½ R² z² − ¼ z⁴) |₀ᴿ

M = kπ (½ R⁴ − ¼ R⁴)

M = ¼ k π R⁴

Next, to find the center of gravity, we use the weighted average:

zG = (∫ z dm) / (∫ dm)

zG = (∫ z dm) / M

We already found M, we just have to evaluate the other integral:

∫ z dm

∫₀ᴿ kπ z² (R² − z²) dz

kπ ∫₀ᴿ (R² z² − z⁴) dz

kπ (⅓ R² z³ − ⅕ z⁵) |₀ᴿ

kπ (⅓ R⁵ − ⅕ R⁵)

²/₁₅ k π R⁵

Plugging in:

zG = (²/₁₅ k π R⁵) / (¼ k π R⁴)

zG = ⁸/₁₅ R

5 0
3 years ago
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