Question

A solid cone of radius 6 cm and height 21 cm is melted and made into 2 spheres of different sizes.The radius of the first sphere is 4 cm , find the radius of the second sphere.

Answer 1

@Michele_Laino

Answer 2

hint:
the volume of the 2 spheres has to be equal to the volume of the starting cone, so we can write this equation:
\[\Large \frac{{4\pi }}{3}{R^3} + \frac{{4\pi }}{3}{x^3} = \frac{{\pi {r^2}h}}{3}\]
where x is the requested radius, R=4 cm, h=21 cm, and r= 6 cm

Answer 3

???

Answer 4

please you have to solve that equation for x

Answer 5

i didnt how you got yhe equation

*the

Answer 6

the sum of the volumes of the 2 spheres, has to be equal to the volume of the starting cone

Answer 7

\[\frac{ 4 }{ 3 }\pi (R^3 + r^3) = \frac{ 1 }{ 3 }\pi(r^2h)\]

Answer 8

not exactly, better is:
\[\Large \frac{{4\pi }}{3}\left( {{R^3} + {x^3}} \right) = \frac{{\pi {r^2}h}}{3}\]
where x is the requested radius, and r is the radius of the starting cone, namely r= 6 cm

Answer 9

3 and pi get cancelled?

Answer 10

yes!

Answer 11

r^h = 4(R^3 + x^3)?

Answer 12

more precisely:
\[\Large 4\left( {{R^3} + {x^3}} \right) = {r^2}h\]

Answer 13

what next?

Answer 14

we have to divide both sides by 4

Answer 15

(R^3 +x^3) = r^2h/4

Answer 16

ok! now we have subtract R^3 at both sides

Answer 17

x^3 = r^2h/4 - R^3

Answer 18

finally we have to take the 3-rd root of both sides

Answer 19

???

Answer 20

you should get this:
\[\Large x = \sqrt[3]{{\frac{{{r^2}h}}{4} - {R^3}}}\]

Answer 21

ok!

Answer 22

\[\Large x = \sqrt[3]{{\frac{{{r^2}h}}{4} - {R^3}}} = \sqrt[3]{{\frac{{{6^2} \cdot 21}}{4} - {4^3}}} = ...?\]

Answer 23

5?

Answer 24

that's right! x= 5 cm

Answer 25

thanks.

Answer 26

:)