Question

32. A solid composed of a cone with hemisphencal base has total surface area of 361.1 sq cm. If the slant height of the cone is 13cm, find the total height of the solid (Take $\pi$= 3.14 )​

Answer 1

I hope it will help you

Answer 2

Answer:

The total height of the solid formed is 17 cm.

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

We have given that,

A solid made up of a cone with hemispherical base has total surface area of 361.1 cm².

The slant height of the cone is 13 cm.

We have to find the height of the total solid.

Let,

TSAˢᵒˡⁱᵈ be total surface area of solid formed

CSAᶜᵒⁿᵉ be curved surface area of the cone

CSAʰᵉᵐⁱˢᵖʰᵉʳᵉ be curved surface area of the hemisphere

Now, we know that,

TSAˢᵒˡⁱᵈ = CSAᶜᵒⁿᵉ + CSAʰᵉᵐⁱˢᵖʰᵉʳᵉ

⇒ TSAˢᵒˡⁱᵈ = ( π r l ) + ( 2 π r² )

⇒ 361.1 = ( 3.14 * r * 13 ) + ( 2 * 3.14 * r² )

⇒ 361.1 = 40.82 r + 6.28 r²

⇒ 6.28 r² + 40.82 r - 361.1 = 0

⇒ 2r² + 13r - 115 = 0 - - [ Dividing each term by 3.14 ]

⇒ 2r² + 23r - 10r - 115 = 0

⇒ r ( 2r + 23 ) - 5 ( 2r + 23 ) = 0

⇒ ( r - 5 ) ( 2r + 23 ) = 0

⇒ r - 5 = 0 or 2r + 23 = 0

⇒ r = 5 or 2r = - 23

⇒ r = 5 or r = - 23 / 2

But, radius i. e. length can't be negative.

∴ r = - 23 / 2 is not acceptable.

∴ r = 5 cm

Now,

In a cone,

l² = r² + h² - - [ Formula ]

⇒ ( 13 )² = ( 5 )² + h²

⇒ 169 = 25 + h²

⇒ h² = 169 - 25

⇒ h² = 144

⇒ h = √144

⇒ h = √( 12 × 12 )

⇒ h = 12 cm

∴ Height of the cone is 12 cm.

Now,

Height of the solid formed = Height of cone + Radius of hemisphere

⇒ Hˢᵒˡⁱᵈ = h + r

⇒ Hˢᵒˡⁱᵈ = 12 + 5

⇒ Hˢᵒˡⁱᵈ = 17 cm

∴ The height of the solid formed is 17 cm.