The radii of the bases of two right circular solid cones of same height are $r_{1}$ and $r_{2}$ respectively. The cones are melted and recast into a solid sphere of radius R. Show that the height of each cone is given by $h = \frac{4 R^{3}}{r_{1}^{2} + r_{2}^{2}}$ .

Open in App

Solution

Let $h$ be the height of each cone and $r_{1}$ and $r_{2}$ be their radius. Then,
Sum of the volumes of two cones = Volume of the sphere
$\frac{1}{3} π r_{1}^{2} h + \frac{1}{3} π r_{2}^{2} h = \frac{4}{3} π R^{3}$

$(r_{1}^{2} + r_{2}^{2}) h = 4 R^{3}$

$h = \frac{4 R^{3}}{r_{1}^{2} + r_{2}^{2}}$

Suggest Corrections

Similar questions

Q. Two solid right circular cones have the same height. The radii of their bases are

r_{1}

and

r_{2}

. They are melted and recast into a cylinder of same height. Show that the radius of the base of the cylinder is

\sqrt{\frac{r_{1}^{2} + r_{2}^{2}}{3}}

The radii of the bases of two solid right circular cones of same height are $r_{1}$ and $r_{2}$ respectively. The cones are melted and recast into a solid sphere of radius R. Find the height of each cone in terms of $r_{1}, r_{2}$ and R.

Q. The radii of two solid metallic spheres are

r_{1}

and

r_{2}

. The spheres are melted together and recast into a solid cone of height

(r_{1} + r_{2})

. Show that the radius of the cone is

2 \sqrt{r_{1}^{2} + r_{2}^{2} - r_{1} r_{2}}

Q. The radius of a solid metallic sphere is

r

. A solid metallic cone of height

h

has base radius

r

. The two are melted together and recast into a solid right circular cone with base radius

r

. Prove that the height of the resulting cone is

4 r + h .

Q. (a) A solid sphere of radius 15 cm is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate the number of cones recast.

(b) A hollow sphere of internal and external radii 6 cm and 8 cm respectively is melted and recast into small cones of base radius 2 cm and height 8 cm. Find the number of cones. [6 MARKS]

Join BYJU'S Learning Program

The radii of the bases of two right circular solid cones of same height are r1 and r2 respectively. The cones are melted and recast into a solid sphere of radius R. Show that the height of each cone is given by h=4R3r21+r22.

Let h be the height of each cone and r1 and r2 be their radius. Then,Sum of the volumes of two cones = Volume of the sphere13πr21h+13πr22h=43πR3(r21+r22)h=4R3h=4R3r21+r22

The radii of the bases of two right circular solid cones of same height are $r_{1}$ and $r_{2}$ respectively. The cones are melted and recast into a solid sphere of radius R. Show that the height of each cone is given by $h = \frac{4 R^{3}}{r_{1}^{2} + r_{2}^{2}}$ .