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alina1380 [7]
3 years ago
10

What is the domain of f(x) = 5^x - 7?

Mathematics
1 answer:
oee [108]3 years ago
5 0
X belongs to all real numbers
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Evaluate the surface integral:S
rjkz [21]
Assuming S does not include the plane z=0, we can parameterize the region in spherical coordinates using

\mathbf r(u,v)=\left\langle3\cos u\sin v,3\sin u\sin v,3\cos v\right\rangle

where 0\le u\le2\pi and 0\le v\le\dfrac\pi/2. We then have

x^2+y^2=9\cos^2u\sin^2v+9\sin^2u\sin^2v=9\sin^2v
(x^2+y^2)=9\sin^2v(3\cos v)=27\sin^2v\cos v

Then the surface integral is equivalent to

\displaystyle\iint_S(x^2+y^2)z\,\mathrm dS=27\int_{u=0}^{u=2\pi}\int_{v=0}^{v=\pi/2}\sin^2v\cos v\left\|\frac{\partial\mathbf r(u,v)}{\partial u}\times \frac{\partial\mathbf r(u,v)}{\partial u}\right\|\,\mathrm dv\,\mathrm du

We have

\dfrac{\partial\mathbf r(u,v)}{\partial u}=\langle-3\sin u\sin v,3\cos u\sin v,0\rangle
\dfrac{\partial\mathbf r(u,v)}{\partial v}=\langle3\cos u\cos v,3\sin u\cos v,-3\sin v\rangle
\implies\dfrac{\partial\mathbf r(u,v)}{\partial u}\times\dfrac{\partial\mathbf r(u,v)}{\partial v}=\langle-9\cos u\sin^2v,-9\sin u\sin^2v,-9\cos v\sin v\rangle
\implies\left\|\dfrac{\partial\mathbf r(u,v)}{\partial u}\times\dfrac{\partial\mathbf r(u,v)}{\partial v}\|=9\sin v

So the surface integral is equivalent to

\displaystyle243\int_{u=0}^{u=2\pi}\int_{v=0}^{v=\pi/2}\sin^3v\cos v\,\mathrm dv\,\mathrm du
=\displaystyle486\pi\int_{v=0}^{v=\pi/2}\sin^3v\cos v\,\mathrm dv
=\displaystyle486\pi\int_{w=0}^{w=1}w^3\,\mathrm dw

where w=\sin v\implies\mathrm dw=\cos v\,\mathrm dv.

=\dfrac{243}2\pi w^4\bigg|_{w=0}^{w=1}
=\dfrac{243}2\pi
4 0
3 years ago
Evaluate 10 + 4(3 + 2) + 5 + (12 ÷ 6).
wolverine [178]
10 + 4*3 + 4*2 + 5 + 2 = 

<span>10 + 12 + 8 +7 = </span>

<span>22 + 8 +7 = </span>

<span>30 + 7 =37</span>
7 0
3 years ago
Read 2 more answers
If a ratio is 4:5, how many parts are there in<br> total?
hjlf
<h2><em><u>♣️</u></em><em><u>A</u></em><em><u>N</u></em><em><u>SWER♣️</u></em></h2>

<h2><em>9</em><em> </em><em>parts </em><em>in </em><em>total</em></h2>

5 0
2 years ago
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Wo cyclists, 72 miles apart, start riding toward each other at the same time. One cycles 2 times as fast as the other. If they m
Trava [24]

Answer:

the faster cyclist was going 18 mph

the slower cyclist was going 9 mph

Step-by-step explanation:

72 ÷ 4 = 18

18 ÷ 2 = 9

3 0
3 years ago
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Mr.Hall stands 200 meters from the base of the water tower. He finds the angle of elevation to the top of the water to be 27 deg
love history [14]

Height of the tower = 101.9 m

Solution:

Given data:

Angle θ = 27°

Adjacent side = 200 m

Opposite side = Height

Let height be taken as h.

The given image shows it is a right triangle.

Using basic trigonometric ratio formulas,

$\tan \theta=\frac{\text { Opposite side of } \theta}{\text { Adjacent side of } \theta}

$\tan27^\circ =\frac{{ h}}{200}

The value of tan 27° = 0.5095

$0.5095 =\frac{{ h}}{200}

Do cross multiplication, we get

0.5095 × 200 = h

101.9 = h

Height of the tower = 101.9 m

8 0
3 years ago
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