Respuesta :
Answer:
The internal diameter of the sphere is 6 cm.
Step-by-step explanation:
Given that, a solid right circular cone of diameter 14 cm and height 8 cm.
The radius of the cone is [tex]=\frac{diameter}{2}[/tex]
[tex]=\frac{14}{2} \ cm[/tex]
= 7 cm.
The volume of the cone is [tex]= \frac13 \pi r^2 h[/tex]
[tex]=(\frac1 3\times \pi \times 7^2\times 8) \ cm^3[/tex]
[tex]=\frac{392}{3}\pi \ cm^3[/tex]
Let the internal radius of the sphere be r.
The external diameter of the sphere is = 10 cm
The external radius of the sphere is(R) = 5 cm
The volume of the sphere is [tex]= \frac43 \pi(R^3-r^3)[/tex]
[tex]=\frac43 \pi (5^3-r^3) \ cm^3[/tex]
The sphere is formed by the solid right circular cone.
∴The volume of the sphere = The volume of the cone
According to the problem,
[tex]\frac43 \pi (5^3-r^3) =\frac{392}{3}\pi[/tex]
[tex]\Rightarrow 4(5^3-r^3)= 392[/tex]
[tex]\Rightarrow 5^3-r^3=\frac{392}{4}[/tex]
[tex]\Rightarrow 5^3-r^3=98[/tex]
[tex]\Rightarrow -r^3=98-125[/tex]
[tex]\Rightarrow-r^3= -27[/tex]
⇒r= 3
The internal radius of the sphere is = 3 cm.
The internal diameter of the given sphere is = (2×3) cm =6 cm.
