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vlabodo [156]
2 years ago
12

Convert the mixed numbers to improper fractions:1. 5 6/72. 8 3/9

Mathematics
1 answer:
finlep [7]2 years ago
5 0
1. 5x72+6 over 72 = 366/72
2. 8x9+3 over 9. = 75/9
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Let f(x,y,z) = ztan-1(y2) i + z3ln(x2 + 1) j + z k. find the flux of f across the part of the paraboloid x2 + y2 + z = 3 that li
Sophie [7]
Consider the closed region V bounded simultaneously by the paraboloid and plane, jointly denoted S. By the divergence theorem,

\displaystyle\iint_S\mathbf f(x,y,z)\cdot\mathrm dS=\iiint_V\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV

And since we have

\nabla\cdot\mathbf f(x,y,z)=1

the volume integral will be much easier to compute. Converting to cylindrical coordinates, we have

\displaystyle\iiint_V\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV=\iiint_V\mathrm dV
=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=1}\int_{z=2}^{z=3-r^2}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta
=\displaystyle2\pi\int_{r=0}^{r=1}r(3-r^2-2)\,\mathrm dr
=\dfrac\pi2

Then the integral over the paraboloid would be the difference of the integral over the total surface and the integral over the disk. Denoting the disk by D, we have

\displaystyle\iint_{S-D}\mathbf f\cdot\mathrm dS=\frac\pi2-\iint_D\mathbf f\cdot\mathrm dS

Parameterize D by

\mathbf s(u,v)=u\cos v\,\mathbf i+u\sin v\,\mathbf j+2\,\mathbf k
\implies\mathbf s_u\times\mathbf s_v=u\,\mathbf k

which would give a unit normal vector of \mathbf k. However, the divergence theorem requires that the closed surface S be oriented with outward-pointing normal vectors, which means we should instead use \mathbf s_v\times\mathbf s_u=-u\,\mathbf k.

Now,

\displaystyle\iint_D\mathbf f\cdot\mathrm dS=\int_{u=0}^{u=1}\int_{v=0}^{v=2\pi}\mathbf f(x(u,v),y(u,v),z(u,v))\cdot(-u\,\mathbf k)\,\mathrm dv\,\mathrm du
=\displaystyle-4\pi\int_{u=0}^{u=1}u\,\mathrm du
=-2\pi

So, the flux over the paraboloid alone is

\displaystyle\iint_{S-D}\mathbf f\cdot\mathrm dS=\frac\pi2-(-2\pi)=\dfrac{5\pi}2
6 0
3 years ago
The diameter of a circle measures 24ft. What is the circumference of the circle? Use 3.14 for pie , and do not round your answer
dexar [7]

Answer:

75.36

Step-by-step explanation:


5 0
3 years ago
10m^2 of tiles cover a pool, the pool is shaped as a square, so all four edges are the same, and the depth is constant but not e
nignag [31]
Assume:

Size of sides = x m
Depth of the pool = y m

Therefore, surface area = x^2+4xy =10 m^2

Then, y = (10-x^2)/(4x)

Now,
Volume (V) = x^2*y = x^2*y =x^2(10-x^2)/4x = (10x-x^3)/4 = 1/4(10x-x^3)
For maximum volume, first derivative of volume function is equal to zero.
That is,
dV/dx =0 = 1/4(10-3x^2)
Then,
1/4(10-3x^2) = 0
10-3x^2 = 0
3x^2=10
x= sqrt (10/3) = 1.826 m
And
y= (10-1.826^2)/(4*1.826) = 0.913 m

Therefore,
V= 1.826^2*0.913 = 3.044 m^3
3 0
3 years ago
Let ∠D be an acute angle such that cosD=0.33. Use a calculator to approximate the measure of ∠D to the nearest tenth of a degree
tia_tia [17]

Answer:

.3

Step-by-step explanation:

.33 rounded equals .3

6 0
3 years ago
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A petrol pump can hold enough petrol to supply 80
liberstina [14]
The answer is 56 cars because there is 2800 litres in the pump so if its 50 litres its going to be 2800/50=56
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2 years ago
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