Please help me with math
1 answer:
The area of triangle BMN is 124.7 square centimeter.
<h3>What are similar triangles?</h3>Similar triangles are triangles that their corresponding angles are equal and the ratio of their corresponding lengths are equal. They are often similar When at least Two angles are equal or The ratio of any corresponding lengths are same.
Analysis:
AB = AC = BC = 50cm
AH = AC/2 = 50/2 = 25cm
From Δ ABH, using Pythagoras theorem to find BH.
=
-
=
-
= 2500 - 625
= 1875
BH = = 43.3cm
ΔBMN and BHC are similar,
so, BM/BH = MN/HC
BM/43.3 = 12/25
25BM = 12 X 43.3
BM = 519.6/25 = 20.78cm
Area of ΔBMN
= 1/2(MN)(BM) = 1/2 x 12 x 20.78 = 124.7 square centimeter
Learn more about similar triangles: brainly.com/question/2644832
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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
