the radius in centimetres of the greatest sphere the can be carried out of a solId cone of radius 9 cm and height 40 cm is
Answers
Step-by-step explanation:
The greatest sphere that can be carved out of a solid cone of radius 9 cm and height 40 cm is :
Slant height of sphere = l²= h²+ r²
l² = 40² +9²
l² = 1681
l = 41 meter
Let the greatest sphere has radius R.
As sphere will touch the base as well as other two sides of the cone.
Line segment PS will be tangent to the sphere at point M.
So, ∠PMO= 90°
As, PR ⊥ QS, ∠PRS = 90°
In Δ PRS, and Δ PMO
m∠PRS = m∠PMO = 90°
∠MPR = ∠SPR [Common Angles]
By AA postulate of Similarity of triangles. We get, ΔPRS ~ ΔPMO
As triangles are similar their sides will be proportional :
\begin{gathered}\frac{PM}{PR}=\frac{MO}{RS}=\frac{PO}{PS}\\\\\frac{R}{9}=\frac{40-R}{41}\\\\\implies 9\cdot (40-R)=41\cdot R\\\\\implies 50\cdot R=360\\\\\bf\implies R=7.20\end{gathered}PRPM=RSMO=PSPO9R=4140−R⟹9⋅(40−R)=41⋅R⟹50⋅R=360⟹R=7.20
Hence, Radius of Greatest sphere = 7.20 meter