Answer:
[tex]\huge{\purple {r= 2\times\sqrt[3]3}}[/tex]
[tex]\huge 2\times \sqrt [3]3 = 2.88[/tex]
Step-by-step explanation:
- For solid iron sphere:
- radius (r) = 2 cm (Given)
- Formula for [tex]V_{sphere} [/tex] is given as:
- [tex]V_{sphere} =\frac{4}{3}\pi r^3[/tex]
- [tex]\implies V_{sphere} =\frac{4}{3}\pi (2)^3[/tex]
- [tex]\implies V_{sphere} =\frac{32}{3}\pi \:cm^3[/tex]
- For cone:
- r : h = 3 : 4 (Given)
- Let r = 3x & h = 4x
- Formula for [tex]V_{cone} [/tex] is given as:
- [tex]V_{cone} =\frac{1}{3}\pi r^2h[/tex]
- [tex]\implies V_{cone} =\frac{1}{3}\pi (3x)^2(4x)[/tex]
- [tex]\implies V_{cone} =\frac{1}{3}\pi (36x^3)[/tex]
- [tex]\implies V_{cone} =12\pi x^3\: cm^3[/tex]
- It is given that: iron sphere is melted and recasted in a solid right circular cone of same volume
- [tex]\implies V_{cone} = V_{sphere}[/tex]
- [tex]\implies 12\cancel{\pi} x^3= \frac{32}{3}\cancel{\pi}[/tex]
- [tex]\implies 12x^3= \frac{32}{3}[/tex]
- [tex]\implies x^3= \frac{32}{36}[/tex]
- [tex]\implies x^3= \frac{8}{9}[/tex]
- [tex]\implies x= \sqrt[3]{\frac{8}{3^2}}[/tex]
- [tex]\implies x={\frac{2}{ \sqrt[3]{3^2}}}[/tex]
- [tex]\because r = 3x [/tex]
- [tex]\implies r=3\times {\frac{2}{ \sqrt[3]{3^2}}}[/tex]
- [tex]\implies r=3\times 2(3)^{-\frac{2}{3}}[/tex]
- [tex]\implies r= 2\times (3)^{1-\frac{2}{3}}[/tex]
- [tex]\implies r= 2\times (3)^{\frac{1}{3}}[/tex]
- [tex]\implies \huge{\purple {r= 2\times\sqrt[3]3}}[/tex]
- Assuming log on both sides, we find:
- [tex]log r = log (2\times \sqrt [3]3)[/tex]
- [tex]log r = log (2\times 3^{\frac{1}{3}})[/tex]
- [tex]log r = log 2+ log 3^{\frac{1}{3}}[/tex]
- [tex]log r = log 2+ \frac{1}{3}log 3[/tex]
- [tex]log r = 0.4600704139[/tex]
- Taking antilog on both sides, we find:
- [tex]antilog(log r )= antilog(0.4600704139)[/tex]
- [tex]\implies r = 2.8844991406[/tex]
- [tex]\implies \huge \red{r = 2.88\: cm}[/tex]
- [tex]\implies 2\times \sqrt [3]3 = 2.88[/tex]
