The diagonal height of a cone with a hemispherical base is 5 cm. If the total surface area of this composite is 103.62cm², find its total height.
Answers
Answer:
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
Given:
The diagonal height of a cone with a hemispherical base is 5 cm.
Total surface area of this composite is 103.62cm²,
To find:
Total height of the cone
Solution:
As we know that the curved surface area of a cone is πrl
And, also the curved surface area of the hemisphere = 2πr²
As per the question, it is given that,
The total surface area of the composite = 103.62 cm²
The diagonal height of the cone, as considered to be l = 5 cm
Let the radius of the hemispherical base be considered as “r” cm.
Now,
The curved surface area of the conical part of the cone = πrl (as per the formula)
= 3.14 * r * 5
= [15.7 r] cm²
Now,
The curved surface area of the hemisphere part of the solid = 2πr² (as per the formula)
= 2 * 3.14 * r²
= [6.28r²] cm².
As we know,
The total surface area of the solid = [C.S.A of the conical part] + [C.S.A of the hemisphere part]
⇒ 103.62 = [ 15.7r] + [6.28r²]
⇒ 6.28r² + 15.7r – 103.62 = 0
⇒ 6.28 r² - 18.84r + 34.54r – 103.62 = 0
⇒ 6.28r (r-3) + 34.54 (r-3) = 0
⇒ (r-3) (6.28r+34.54) = 0
⇒ r = 3 cm [neglecting the negative value as it is not possible]
Again,
Let the height of the cone be considered as “h” cm.
We know, l² = h² + r² ( as per the formula of the slant height of a cone is given)
Hence, substituting the value of r and l in the formula,
⇒h = √[5² – 3²]
⇒ h = √16
⇒ h = 4 cm
Now, the total height of the cone is given as = [height of the cone] + [radius of the hemisphere base]
= h + r
= 4 + 3
= 7 cm
Thus, the total height of the cone is 7 cm.