How to prove formula for volume of a sphere ?

The volume of radius is   × π × radius³  Proved

Explanation:

Given as :

We know that volume of sphere is v = × π × radius³

Or, v =   × π × r³

Let prove the volume of sphere

So, From the figure of sphere

At the height of z , there is shaded disk with radius x

Let Find the area of triangle with side x , z , r

From Pythagorean theorem

x² + z² = r²

Or, x² = r² -  z²

Or, x =

Now, Area of shaded disk = Area = π × x²

Where x is the radius of disk

Or, Area of shaded disk = π × () ²

∴ Area of shaded disk = π × (r² - z²)

Again

If we calculate the area of all horizontal disk, we can get the volume of sphere

So, we simply integrate the area of all disk from - r to + r

i.e volume =

Or, v = -

Or, v = π r² (r + r) -  π

Or, v = π r² (r + r) - π

Or, v = 2πr³ - π

Or, v =  2πr³ ()

Or, v = 2πr³ ×

∴ v =   × π × r³

Hence, The volume of radius is   × π × radius³  Proved . Answer

a solenoid is created by wrapping a l = 90 m long wire around a hollow tube of diameter d = 4.5 cm.   the wire diameter is d = 0.9 mm.   the solenoid wire is then connected to a power supply so that a current of i = 9 a flows through the wire.

randomized variablesl = 90 m

d = 4.5 cm

d = 0.9 mm

i = 9 a                                          

write an expression for the number of turns, n, in the solenoid. you do not need to take into account the diameter of the wire in this calculation.

calculate the number of turns, n, in the solenoid.

write an expression for the length of the solenoid (l2) in terms of the diameter of the hollow tube d.

calculate the length of the solenoid (l2)   in meters.

calculate the magnitude of the magnetic field at the center of the solenoid in teslas.

explanation:

that would be because of both gravity and friction. so     c