## anonymous one year ago 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.

1. anonymous

@Michele_Laino

2. Michele_Laino

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

3. anonymous

???

4. Michele_Laino

please you have to solve that equation for x

5. anonymous

i didnt how you got yhe equation *the

6. Michele_Laino

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

7. anonymous

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

8. Michele_Laino

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

9. anonymous

3 and pi get cancelled?

10. Michele_Laino

yes!

11. anonymous

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

12. Michele_Laino

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

13. anonymous

what next?

14. Michele_Laino

we have to divide both sides by 4

15. anonymous

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

16. Michele_Laino

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

17. anonymous

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

18. Michele_Laino

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

19. anonymous

???

20. Michele_Laino

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

21. anonymous

ok!

22. Michele_Laino

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

23. anonymous

5?

24. Michele_Laino

that's right! x= 5 cm

25. anonymous

thanks.

26. Michele_Laino

:)