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Dima020 [189]
3 years ago
13

G find the area of the surface over the given region. use a computer algebra system to verify your results. the sphere r(u,v) =

a sin u cos v i + a sin u sin v j + a cos u k, where 0 ≤ u ≤ π and 0 ≤ v ≤ 2π
Mathematics
1 answer:
Svetach [21]3 years ago
6 0
Presumably you should be doing this using calculus methods, namely computing the surface integral along \mathbf r(u,v).

But since \mathbf r(u,v) describes a sphere, we can simply recall that the surface area of a sphere of radius a is 4\pi a^2.

In calculus terms, we would first find an expression for the surface element, which is given by

\displaystyle\iint_S\mathrm dS=\iint_S\left\|\frac{\partial\mathbf r}{\partial u}\times\frac{\partial\mathbf r}{\partial v}\right\|\,\mathrm du\,\mathrm dv

\dfrac{\partial\mathbf r}{\partial u}=a\cos u\cos v\,\mathbf i+a\cos u\sin v\,\mathbf j-a\sin u\,\mathbf k
\dfrac{\partial\mathbf r}{\partial v}=-a\sin u\sin v\,\mathbf i+a\sin u\cos v\,\mathbf j
\implies\dfrac{\partial\mathbf r}{\partial u}\times\dfrac{\partial\mathbf r}{\partial v}=a^2\sin^2u\cos v\,\mathbf i+a^2\sin^2u\sin v\,\mathbf j+a^2\sin u\cos u\,\mathbf k
\implies\left\|\dfrac{\partial\mathbf r}{\partial u}\times\dfrac{\partial\mathbf r}{\partial v}\right\|=a^2\sin u

So the area of the surface is

\displaystyle\iint_S\mathrm dS=\int_{u=0}^{u=\pi}\int_{v=0}^{v=2\pi}a^2\sin u\,\mathrm dv\,\mathrm du=2\pi a^2\int_{u=0}^{u=\pi}\sin u
=-2\pi a^2(\cos\pi-\cos 0)
=-2\pi a^2(-1-1)
=4\pi a^2

as expected.
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The solid shape is made from a cylinder and a hemisphere. The radius of the cylinder is equal to the radius of the hemisphere. The cylinder has a height of 10 cm. The curved surface area of the hemisphere is 32π cm2.

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HELP!!<br> Full Question in picture
drek231 [11]

a. The value of x is 7

b. The measure of ∠1 is 99°

<h3>Calculating angles </h3>

From the question, we are to solve for x

From the given diagram, we can write that

m∠NMQ + m∠MQN + m∠QNM = 180° (<em>Sum of angles in a triangle</em>)

From the given information,

m∠NMQ = 5x +19

m∠MQN = 8x -11

m∠QNM = 11x + 4

Then,

5x + 19 + 8x -11 + 11x + 4 = 180

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5x + 8x + 11x = 180 - 19 + 11 - 4

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b.
Measure of ∠1 + m∠QNM = 180° (<em>Sum of angles on a straight line</em>)

∴ Measure of ∠1 = 180° - m∠QNM

But m∠QNM = 11x + 4

∴ m∠QNM = 11(7) + 4

m∠QNM = 77 + 4

m∠QNM = 81°

Then,

Measure of ∠1 = 180° - 81°

Measure of ∠1 = 99°

Hence, the measure of ∠1 is 99°

Learn more on Calculating angles here: brainly.com/question/25716982

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