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Dima020 [189]
2 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]2 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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A 97% confidence interval for the mean of a population is to be constructed and must be accurate to within 0.3 unit. A prelimina
ivann1987 [24]

Answer:

The sample size will be "102.5494 \sim 102". A further explanation is given below.

Step-by-step explanation:

Let the sample size will be:

= n

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As we know,

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On putting the estimated values, we get

⇒    =(2.17\times \frac{1.4}{0 .3})^2

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⇒    =102.5494 \sim 102

4 0
3 years ago
Find the volume of the garden shed
Leno4ka [110]

Answer:

47.3 m³

Step-by-step explanation:

The garden shed is made up of a rectangular prism and a pyramid.

<h3><u>Volume of a rectangular prism</u></h3>

\begin{aligned}\textsf{Volume of a rectangular prism}&=\sf width \times length \times height\\&=4 \times 4 \times 2\\&=16 \times 2\\&=32\;\; \sf m^3\end{aligned}

<h3><u>Volume of a pyramid</u></h3>

<u />

From inspection of the given diagram, the slant height of the pyramid is 3.5 m.  

Calculate the perpendicular height of the pyramid using Pythagoras Theorem:

\begin{aligned}\implies a^2+b^2&=c^2\\2^2+h^2&=3.5^2\\h^2&=8.25\\h&=\sqrt{8.25}\;\; \sf m\end{aligned}

Therefore:

\begin{aligned}\textsf{Volume of a pyramid}&=\dfrac{\sf length \times width \times height}{3}\\\\&=\dfrac{\sf 4 \times 4 \times \sqrt{8.25}}{3}\\\\& = 15.31883372\;\; \sf m^3\end{aligned}

<h3><u>Volume of the garden shed</u></h3>

\begin{aligned}\implies \textsf{Volume of shed}&=\textsf{Volume of rectangular prism}+\textsf{Volume of pyramid}\\&=32+15.31883372\\& = 47.3\;\; \sf m^3\;(nearest\;tenth)\end{aligned}

6 0
1 year ago
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Answer:

What error?

Step-by-step explanation

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2 years ago
How many liters of 70% alcohol solution and 30% alcohol solution must be mixed to obtain 8 liters of 60% alcohol solution?
vfiekz [6]
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2 years ago
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andreev551 [17]

Answer: x = 11

Step-by-step explanation:

same as the last question, set them as equal and solve. here's my work:

<em>5x - 1 = 4x + 10</em>

<em>5x - 1 (+ 1) = 4x + 10 (+ 1)</em>

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2 years ago
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