Answer:

If a solid is composed of a cone with a hemispherical base WHOSE area is 361.1 cm^2 and slant height is 13 cm then the total height of the solid is 17 cm.

Step-by-step EXPLANATION:

Required FORMULAS:

Curved Surface area of the Cone = πrl

Curved Surface area of the hemisphere = 2πr²  

It is given that,

A solid is in the shape of a cone with hemispherical base

The total surface area of the solid = 361.1 cm²

The slant height of the cone, l = 13 cm

Step 1:

Let the radius of the hemispherical base be denoted as “r” cm.

Now,

The curved surface area of the CONICAL part of the solid = πrl = 3.14 * r * 13 = [40.82 r] cm²

And,

The curved surface area of the hemisphere part of the solid = 2πr² = 2 * 3.14 * r² = [6.28r²] cm².

We KNOW that,

The total surface area of the solid = [C.S.A of the conical part] + [C.S.A of the hemisphere part]

⇒ 361.1 = [40.82 r] + [6.28r²]  

⇒ 6.28r² + 40.82r – 361.1 = 0

⇒ r² + 6.5r – 57.5 = 0

⇒ r² + 11.5r – 5r – 57.5 = 0

⇒ r(r+11.5) – 5(r+11.5) = 0

⇒ (r+11.5)(r-5) = 0

⇒ r = 5 cm …… [neglecting the negative value]

Step 2:

Let the height of the cone be denoted as “h” cm.

We know the formula of the slant height of a cone is given by,

l² = h² + r²  

Substituting the value of l and r in the formula, we get

h = √[13² – 5²]

⇒ h = √[144]  

⇒ h = 12 cm

Thus,  

The total height of the solid is given by,

= [height of the cone] + [radius of the hemisphere base]

= h + r

= 12 + 5

= 17 cm