Question

distance of centre of mass of a solid uniform cone from its vertex is z₀. if radius of its base is R and its height is h, then z₀ = ?​

Answer 1

Step-by-step explanation:

The height of a cone is 30 cm. From its topside a small cone is cut by a plane parallel to its base. If volume of smaller cone is 1/27 of the given cone, then at what height it is cut from its base?

Answer:

Given height of cone, H= 30 cm

Let radius of the cone be R.

Let cone of height h be cut off from the top of the given cone and radius be r.

Consider ΔAPC and ΔAQE,

PC || QE

∴ ΔAPC ~ ΔAQE

⇒ AP/PQ = PC/QE

⇒ h/H = r/R … (1)

Given, Volume of smaller cone = 1/27 (Volume of given cone)

So, Volume of cone ABC = 1/27 (Volume of cone ADE)

Volume of cone ABC / Volume of cone ADE = 1/27

We know that volume of cone = πr2h/3

So,

⇒ (πr2h/3) / (πR2H/3) = 1/27

⇒ (r/R)2(h/H) = 1/27

From (1),

⇒ (h/H)2(h/H) = 1/27

⇒ (h/H)3 = 1/27

⇒ h/H = 1/3

⇒ h = (1/3) H

⇒ h = (1/3) (30)

∴ h = 10 cm

Now, PQ = H – h = 30 – 10

∴ PQ = 20 cm

Ans. The height at which the cone is cut from its base is 20 cm.