Evaluate m) (x) for x = -3.<br> =<br> I<br> m (-3) =

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

2 years ago

Pls help with this problem

madam [21]

Step-by-step explanation:

every triangle inscribed into a circle with the baseline being a diameter of the circle must be a right-angled triangle.

therefore,

angle XVY = 90°.

for VY we can use Pythagoras

c² = a² + b²

with c being the Hypotenuse (the line opposite of the 90° angle, in our case XY).

XY = 2×ZY = 2×17 = 34.

so,

34² = 30² + VY²

VY² = 34² - 30² = 1156 - 900 = 256

VY = 16

6 0

1 year ago

Help me with this easy problem If you answer correctly you will get brainliest and I wll draw you something. whaterver you like!

hjlf

Answer: 620

Step-by-step explanation:

$40 x 5.5 hrs = $220 This means that she earned $220 tutoring the first student.

$40 x 10 hrs = $400 She earned $400 tutoring the second student.

$220 + $400 = $620 She earned $620 in total that week.

7 0

1 year ago

Read 2 more answers

The function H(t) = −16t2 + 75t + 25 shows the height H(t), in feet, of a baseball after t seconds. Part A: After how many secon

nlexa [21]

Step-by-step explanation:

Given that,

The function that shows the height of a baseball after t seconds is given by :

$H(t)=-16t^2+75t+25$ ....(1)

Part (A).

When the baseball hits the ground,

H(t) = 0

So,

$-16t^2+75t+25=0\\\\t=5\ s\ and\ t=-0.31\ s$

After 5 seconds the baseball will hit the ground.

Pat (B)

For maximum height, put dH/dt = 0

So,

$\dfrac{d}{dt}(-16t^2+75t+25)=0\\\\-32t+75=0\\\\t=2.34\ s$

Put t = 2.34 s in equation (1).

$H(t)=-16(2.34)^2+75(2.34)+25\\\\H(t)=112.89\ m$

Hence, this is the required solution.

8 0

2 years ago

What is the general form to find square roots

Effectus [21]

Answer:

see below (I hope this helps!)

Step-by-step explanation:

Square roots are usually in the form a√b where a and b are any rational number.

4 0

2 years ago

Evaluate m) (x) for x = -3. = I m (-3) =