A fraction that is less than 5/8 but with the same denominator

Find the surface area of x^2+y^2+z^2=9 that lies above the cone z= sqrt(x^@+y^2)

Mashcka [7]

The cone equation gives

$z=\sqrt{x^2+y^2}\implies z^2=x^2+y^2$

which means that the intersection of the cone and sphere occurs at

$x^2+y^2+(x^2+y^2)=9\implies x^2+y^2=\dfrac92$

i.e. along the vertical cylinder of radius $\dfrac3{\sqrt2}$ when $z=\dfrac3{\sqrt2}$ .

We can parameterize the spherical cap in spherical coordinates by

$\mathbf r(\theta,\varphi)=\langle3\cos\theta\sin\varphi,3\sin\theta\sin\varphi,3\cos\varphi\right\rangle$

where $0\le\theta\le2\pi$ and $0\le\varphi\le\dfrac\pi4$ , which follows from the fact that the radius of the sphere is 3 and the height at which the sphere and cone intersect is $\dfrac3{\sqrt2}$ . So the angle between the vertical line through the origin and any line through the origin normal to the sphere along the cone's surface is

$\varphi=\cos^{-1}\left(\dfrac{\frac3{\sqrt2}}3\right)=\cos^{-1}\left(\dfrac1{\sqrt2}\right)=\dfrac\pi4$

Now the surface area of the cap is given by the surface integral,

$\displaystyle\iint_{\text{cap}}\mathrm dS=\int_{\theta=0}^{\theta=2\pi}\int_{\varphi=0}^{\varphi=\pi/4}\|\mathbf r_u\times\mathbf r_v\|\,\mathrm dv\,\mathrm du$
$=\displaystyle\int_{u=0}^{u=2\pi}\int_{\varphi=0}^{\varphi=\pi/4}9\sin v\,\mathrm dv\,\mathrm du$
$=-18\pi\cos v\bigg|_{v=0}^{v=\pi/4}$
$=18\pi\left(1-\dfrac1{\sqrt2}\right)$
$=9(2-\sqrt2)\pi$

3 0

3 years ago

How to write the slope intercept equation for points of (6,0) and (0,-5)?

Vadim26 [7]

Answer:

$y=\frac{5}{6} x -5$

Step-by-step explanation:

Hi there!

We are given the points (6,0) and (0, -5), and we want to write the equation of the line containing those points in slope-intercept form

Slope-intercept form can be written as y=mx+b, where m is the slope and b is the y intercept

First, we need to find the slope of the line

The formula for the slope can be written as $\frac{y_2-y_1}{x_2-x_1}$ , where $(x_1, y_1)$ and $(x_2, y_2)$ are points

We have everything we need to find the slope, but let's label the points to avoid confusion

$x_1=6\\y_1=0\\x_2=0\\y_2=-5$

Now substitute these values into the formula

m= $\frac{y_2-y_1}{x_2-x_1}$

m= $\frac{-5-0}{0-6}$

Subtract

m= $\frac{-5}{-6}$

Simplify

m= $\frac{5}{6}$

The slope of the line is 5/6

Let's plug this value into the formula for the equation of the line in slope-intercept form.

We substitute 5/6 for m in y=mx+b:
y= $\frac{5}{6}x+b$

Now we need to find b

As stated before, b is the y intercept, which is the value where the line hits the y axis. The value of x at the y intercept is 0.

One of the points we were given is actually the y intercept; that point is (0, -5); notice how the value of x in this point is 0

The value of b is the value of y in this point, which is -5 in this case.

Substitute -5 as b in the formula.

$y=\frac{5}{6} x -5$

Hope this helps!

See more on this subject here (n.b. the answer uses a different way of solving): brainly.com/question/20891204

8 0

2 years ago

Freddie ran for 5 days in a row. The line plot shows how much miles he ran each day. how far did he ran in total? Be sure to red

Blababa [14]

Answer:

8/4

Step-by-step explanation:

7 0

2 years ago

What function is 2x+3y=12

sergeinik [125]

The answer to your question is alphabetical order

7 0

3 years ago

PLEASE HELP ME. I WILL GIVE BRAINLIEST TO THE FIRST PERSON WHO ANSWERS THIS QUESTION. I NEED AN ANSWER ASAP. FOLLOW ALL THE DIRE

Korolek [52]

Answer:

x=3*2 1/3

Step-by-step explanation:

In order to solve this non-linear equation, we need to move all the terms to the left side.

In our example,

- term 3, will be moved to the right side.

Notice that a term changes sign when it 'moves' from one side of the equation to the other.

(x^3 + 3) + (-3) = 57 + (-3)

We need to get rid of expression parentheses.

If there is a negative sign in front of it, each term within the expression changes sign.

Otherwise, the expression remains unchanged.

In our example, there are no negative expressions.

x^3 + 3 - 3 = 57 - 3

4 0

3 years ago