Answer:
Step-by-step explanation:
Given the surface
G(x, y, z) =2z²
Over the hemisphere
x² + y² + z² = 36. For z≥0
Using polar coordinate
x=sin Φ cos θ,
y = sin Φ sin θ,
z = cos Φ
0 ≤ Φ ≤ π/2, 0 ≤ θ ≤ 2π
Therefore
r(Φ, θ) = sin Φ cos θ i + sin Φ sin θ j + cos Φ k
Also, dS= |rθ×rΦ|= sinΦ
dS=sinΦdΦdθ
Then we want to compute the volume integral of
∫ ∫ₛ G(x, y, z) dS
G(x, y, z) =2z²
Therefore in polar forms
G(x, y, z) =2(cos Φ)²
G(x, y, z) = 2cos²Φ
Given that dS=sinΦdΦdθ
∫ ∫ₛ G(x, y, z) dS
∫ ∫ 2cos²ΦsinΦdΦdθ at 0 ≤ Φ ≤ π/2,
0 ≤ θ ≤ 2π
∫ 2cos²ΦsinΦ •θdΦ from 0 ≤ θ ≤ 2π
2∫cos²ΦsinΦ •(2π-0)dΦ
4π∫ cos²ΦsinΦ dΦ from 0 ≤ Φ ≤ π/2
Let U = cosΦ
dU/dΦ =-sinΦ
-dU/sinΦ =dΦ
4π∫ U²sinΦ(-dU/sinΦ) 0 ≤ Φ ≤ π/2
-4π∫ U² dU
-4π U³/3, then U=cosΦ
[-4πcos³Φ / 3 ] from 0 ≤ Φ ≤ π/2
[-4π cos³(π/2)/3 - [-4π cos³(0)/3]
0+4π/3
4π/3
4π/3 unit²
Answer:
A. M= 20
B. r = 6
Step-by-step explanation:
M=kr³
where k is the constant
M= 160 , r 4
160= k(4)³
k = 160/4³= 2.5
A. when r= 2 , M= kr³
M= 2.5(2)³ = 20
B. when M= 540, find r
M= kr³
r³ = M/k
r³ = 540/2.5
r³= 216
r = 6
Answer: 30
Step-by-step explanation:
I’m big smart
Step-by-step explanation:
1/10 is equivalent to saying 10/100.
Adding these together would look like:
10/100 + 6/100 = 16/100 (cannot be simplified)
a/b = 16/100
Answer: Let x = the measure of the larger arc and
let y = the measure of the smaller arc which is closest to the angle’s vertex. We know that x + y = 180 because that is given.
The secant-secant angle has a measure that is half the difference of the intercepts arcs. That means (1/2)(x - y) = 32 for this problem
We just need to solve the system
x + y = 180 and x - y = 64
Adding the equations you get 2x = 244 so x = 122 degrees and y = 58 degrees
Step-by-step explanation:
