Answer:
The volume of the solid is 243![\sqrt{2} \ \pi](https://tex.z-dn.net/?f=%5Csqrt%7B2%7D%20%5C%20%5Cpi)
Step-by-step explanation:
From the information given:
BY applying sphere coordinates:
0 ≤ x² + y² + z² ≤ 81
0 ≤ ρ² ≤ 81
0 ≤ ρ ≤ 9
The intersection that takes place in the sphere and the cone is:
![x^2 +y^2 ( \sqrt{x^2 +y^2 })^2 = 81](https://tex.z-dn.net/?f=x%5E2%20%2By%5E2%20%28%20%5Csqrt%7Bx%5E2%20%2By%5E2%20%7D%29%5E2%20%20%3D%2081)
![2(x^2 + y^2) =81](https://tex.z-dn.net/?f=2%28x%5E2%20%2B%20y%5E2%29%20%3D81)
![x^2 +y^2 = \dfrac{81}{2}](https://tex.z-dn.net/?f=x%5E2%20%2By%5E2%20%3D%20%5Cdfrac%7B81%7D%7B2%7D)
Thus; the region bounded is: 0 ≤ θ ≤ 2π
This implies that:
![z = \sqrt{x^2+y^2}](https://tex.z-dn.net/?f=z%20%3D%20%5Csqrt%7Bx%5E2%2By%5E2%7D)
ρcosФ = ρsinФ
tanФ = 1
Ф = π/4
Similarly; in the X-Y plane;
z = 0
ρcosФ = 0
cosФ = 0
Ф = π/2
So here; ![\dfrac{\pi}{4} \leq \phi \le \dfrac{\pi}{2}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpi%7D%7B4%7D%20%5Cleq%20%5Cphi%20%5Cle%20%5Cdfrac%7B%5Cpi%7D%7B2%7D)
Thus, volume: ![V = \iiint_E \ d V = \int \limits^{\pi/2}_{\pi/4} \int \limits ^{2\pi}_{0} \int \limits^9_0 \rho ^2 \ sin \phi \ d\rho \ d \theta \ d \phi](https://tex.z-dn.net/?f=V%20%20%3D%20%5Ciiint_E%20%5C%20d%20V%20%3D%20%5Cint%20%5Climits%5E%7B%5Cpi%2F2%7D_%7B%5Cpi%2F4%7D%20%20%5Cint%20%5Climits%20%5E%7B2%5Cpi%7D_%7B0%7D%20%5Cint%20%5Climits%5E9_0%20%5Crho%20%20%20%5E2%20%5C%20sin%20%5Cphi%20%5C%20d%5Crho%20%5C%20%20%20d%20%5Ctheta%20%5C%20%20d%20%5Cphi)
![V = \int \limits^{\pi/2}_{\pi/4} \ sin \phi \ d \phi \int \limits ^{2\pi}_{0} d \theta \int \limits^9_0 \rho ^2 d\rho](https://tex.z-dn.net/?f=V%20%20%3D%20%5Cint%20%5Climits%5E%7B%5Cpi%2F2%7D_%7B%5Cpi%2F4%7D%20%5C%20sin%20%5Cphi%20%20%5C%20d%20%5Cphi%20%20%5Cint%20%5Climits%20%5E%7B2%5Cpi%7D_%7B0%7D%20d%20%5Ctheta%20%5Cint%20%5Climits%5E9_0%20%5Crho%20%20%20%5E2%20d%5Crho)
![V = \bigg [-cos \phi \bigg]^{\pi/2}_{\pi/4} \bigg [\theta \bigg]^{2 \pi}_{0} \bigg [\dfrac{\rho^3}{3} \bigg ]^{9}_{0}](https://tex.z-dn.net/?f=V%20%3D%20%5Cbigg%20%5B-cos%20%5Cphi%20%20%5Cbigg%5D%5E%7B%5Cpi%2F2%7D_%7B%5Cpi%2F4%7D%20%20%5Cbigg%20%5B%5Ctheta%20%20%5Cbigg%5D%5E%7B2%20%5Cpi%7D_%7B0%7D%20%5Cbigg%20%5B%5Cdfrac%7B%5Crho%5E3%7D%7B3%7D%20%20%5Cbigg%20%5D%5E%7B9%7D_%7B0%7D)
![V = [ -0+ \dfrac{1}{\sqrt{2}}][2 \pi -0] [\dfrac{9^3}{3}- 0 ]](https://tex.z-dn.net/?f=V%20%3D%20%5B%20-0%2B%20%5Cdfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%5D%5B2%20%5Cpi%20-0%5D%20%5B%5Cdfrac%7B9%5E3%7D%7B3%7D-%200%20%5D)
V = 243![\sqrt{2} \ \pi](https://tex.z-dn.net/?f=%5Csqrt%7B2%7D%20%5C%20%5Cpi)
9514 1404 393
Answer:
84 in
Step-by-step explanation:
The Pythagorean theorem is used to find the length 'b'.
b^2 +12^2 = 37^2
b = √(37^2 -12^2) = √1225 = 35
Then the perimeter is the sum of side lengths:
P = a + b + c = 12 + 35 + 37
P = 84 . . . inches
Density is a property of a material which describes the mass of a material per unit volume. It can be calculated by dividing the mass of the object with its volume. We calculate as follows:
Density = mass / volume = 400 / (8^3) = <span>0.78125 g/cm^3</span>
Hi,
I believe this is Italian. Translated, the sentence reads:
<span>How is the area of the square?
Faith xoxo</span>