Ice-Cream Cone problem - Volume in Spherical Coord

[masterchiefo]

Homework Statement

S is the sphere of equation x² + y² + z² = 10z and C the cone of equation
z= sqrt(3*( x² + y²)) . The axes are measured
centimeters.
R of sphere = 5
D = 10

Total height is 10 cm

Illustrate the solid E bounded by the C cone and the sphere S and calculate its volume using the details
Spherical.

Homework Equations

The Attempt at a Solution

To find phi:
x=0
z= sqrt(3*( x² + y²))
z= sqrt(3*( 0² + y²))
z= y*sqrt(3)

y= 3 <== I picked 3, could of picked any number, its just to find the Z and then find the angle.
z= 3*sqrt(3)

tan(phi)=3/(3*sqrt(3))
phi = pi/6 ===30degree

Theta = 2*pi

$=integral
2*pi$0 pi/6$0 10*cos(phi)$0 p2*sin(phi) dp dphi dtheta

229.074cm³

 

[LCKurtz]

I didn't check the value of your integral but it looks like it is set up correctly. Why give a decimal instead of exact answer?

 

[masterchiefo]

I didn't check the value of your integral but it looks like it is set up correctly. Why give a decimal instead of exact answer?

I am used to give decimal as my teacher always ask for that.

 

[LCKurtz]

I didn't check the value of your integral but it looks like it is set up correctly. Why give a decimal instead of exact answer?

I am used to give decimal as my teacher always ask for that.

Nevertheless, what you should routinely do is give the exact answer if it is practical, and at the very end put the decimal approximation. For example, if the exact answer is $\frac {\sqrt 3} 5$ you could write it like this:
$$\text{Your Variable }=\frac {\sqrt 3} 5 \approx .3464$$That would be correct, good form, and would satisfy your teacher.