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3 years ago

If y = 4x + 0.5, find y when x =1.5

Mathematics

1 answer:

inn [45]3 years ago

8 0

Answer:

= 6.5

Step-by-step explanation:

You have to plug in 1.5 for x

4(1.5) + 0.5 = 6.5

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Use the Divergence Theorem to evaluate S F · dS, where F(x, y, z) = z2xi + y3 3 + sin z j + (x2z + y2)k and S is the top half of

GenaCL600 [577]

Close off the hemisphere $S$ by attaching to it the disk $D$ of radius 3 centered at the origin in the plane $z=0$ . By the divergence theorem, we have

$\displaystyle\iint_{S\cup D}\vec F(x,y,z)\cdot\mathrm d\vec S=\iiint_R\mathrm{div}\vec F(x,y,z)\,\mathrm dV$

where $R$ is the interior of the joined surfaces $S\cup D$ .

Compute the divergence of $\vec F$ :

$\mathrm{div}\vec F(x,y,z)=\dfrac{\partial(xz^2)}{\partial x}+\dfrac{\partial\left(\frac{y^3}3+\sin z\right)}{\partial y}+\dfrac{\partial(x^2z+y^2)}{\partial k}=z^2+y^2+x^2$

Compute the integral of the divergence over $R$ . Easily done by converting to cylindrical or spherical coordinates. I'll do the latter:

$\begin{cases}x(\rho,\theta,\varphi)=\rho\cos\theta\sin\varphi\\y(\rho,\theta,\varphi)=\rho\sin\theta\sin\varphi\\z(\rho,\theta,\varphi)=\rho\cos\varphi\end{cases}\implies\begin{cases}x^2+y^2+z^2=\rho^2\\\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi\end{cases}$

So the volume integral is

$\displaystyle\iiint_Rx^2+y^2+z^2\,\mathrm dV=\int_0^{\pi/2}\int_0^{2\pi}\int_0^3\rho^4\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\frac{486\pi}5$

From this we need to subtract the contribution of

$\displaystyle\iint_D\vec F(x,y,z)\cdot\mathrm d\vec S$

that is, the integral of $\vec F$ over the disk, oriented downward. Since $z=0$ in $D$ , we have

$\vec F(x,y,0)=\dfrac{y^3}3\,\vec\jmath+y^2\,\vec k$

Parameterize $D$ by

$\vec r(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath$

where $0\le u\le 3$ and $0\le v\le2\pi$ . Take the normal vector to be

$\dfrac{\partial\vec r}{\partial v}\times\dfrac{\partial\vec r}{\partial u}=-u\,\vec k$

Then taking the dot product of $\vec F$ with the normal vector gives

$\vec F(x(u,v),y(u,v),0)\cdot(-u\,\vec k)=-y(u,v)^2u=-u^3\sin^2v$

So the contribution of integrating $\vec F$ over $D$ is

$\displaystyle\int_0^{2\pi}\int_0^3-u^3\sin^2v\,\mathrm du\,\mathrm dv=-\frac{81\pi}4$

and the value of the integral we want is

(integral of divergence of F) - (integral over D) = integral over S

==> 486π/5 - (-81π/4) = 2349π/20

5 0

3 years ago

Express each of the following quadratic functions in the form of f(x) = a (x - h)²+ k.Then,state the minimum or maximum value,ax

Ivan

Given:

The function is:

$f(x)=-2x^2+7x+4$

To find:

Express the quadratic equation in the form of $f(x)=a(x-h)^2+k$ , then state the minimum or maximum value,axis of symmetry and minimum or maximum point.

Solution:

The vertex form of a quadratic function is:

$f(x)=a(x-h)^2+k$ ...(i)

Where, a is a constant, (h,k) is the vertex and x=h is the axis of symmetry.

We have,

$f(x)=-2x^2+7x+4$

It can be written as:

$f(x)=-2\left(x^2-3.5x\right)+4$

Adding and subtracting square of half of coefficient of x inside the parenthesis, we get

$f(x)=-2\left(x^2-3.5x+(\dfrac{3.5}{2})^2-(\dfrac{3.5}{2})^2\right)+4$

$f(x)=-2\left(x^2-3.5x+(1.75)^2\right)-2\left(-(1.75)^2\right)+4$

$f(x)=-2\left(x-1.75\right)^2+2(3.0625)+4$

$f(x)=-2\left(x-1.75\right)^2+6.125+4$

$f(x)=-2\left(x-1.75\right)^2+10.125$ ...(ii)

On comparing (i) and (ii), we get

$a=-2,h=1.75,k=10.125$

Here, a is negative, the given function represents a downward parabola and its vertex is the point of maxima.

Maximum value = 10.125

Axis of symmetry : $x=1.75$

Maximum point = (1.75,10.125)

Therefore, the vertex form of the given function is $f(x)=-2\left(x-1.75\right)^2+10.125$ , the maximum value is 10.125, the axis of symmetry is $x=1.75$ and the maximum point is (1.75,10.125).

8 0

3 years ago

What is the decimal multiplier to increase by 99%? need it now

krek1111 [17]

Answer:

To work out the multiplier, first add or subtract the percentage from 100, then convert to a decimal. Example: we want to add 20% to £110. To work out the multiplier, add 20 to 100, to get 120, and then change it to a decimal (divide by 100) to get 1.2.

meiabatten191 helped me out on this question too.

3 0

3 years ago

Can someone please explain to me how to get this answer?

SIZIF [17.4K]

Answers:
A = 120
b = 45.0
c = 33.2

Side Note: only one triangle is possible
See attached for a visual. I used GeoGebra to draw the triangle.

-------------------------------------------------------------
-------------------------------------------------------------

Explanation:

We are given the following information
B = 35
C = 25
a = 68

We need to find the following
A, b, c

where the lower case letters represent the side lengths; the upper case letters are the angles. The angles are opposite their corresponding sides. For instance, side lowercase b is opposite angle uppercase B. The other letters are positioned the same way.

-----------------------

First use the idea that for any triangle, the three angles (A,B,C) must add to 180 degrees
A+B+C = 180
Replace B and C with 35 and 25. Solve for angle A
A+35+25 = 180
A+60 = 180
A+60-60 = 180-60
A = 120

Now that we know the three angles A = 120, B = 35, C = 25, we can find the missing sides 'b' and 'c'

-----------------------

We will use the law of sines to find side b
sin(A)/a = sin(B)/b
sin(120)/68 = sin(35)/b
b*sin(120) = 68*sin(35) <<--- cross multiply
b = 68*sin(35)/sin(120)
b = 45.037013350222 <<--- use a calculator; this value is approximate
b = 45.0 <<--- round to the nearest tenth

-----------------------

Do the same for side c
sin(A)/a = sin(C)/c
sin(120)/68 = sin(25)/c
c*sin(120) = 68*sin(25) <<--- cross multiply
c = 68*sin(25)/sin(120)
c = 33.1838323365404 <<--- use a calculator; this value is approximate
c = 33.2 <<--- round to the nearest tenth