The total surface area of a right circular cone of slant height 13 cm is 90 π cm². Calcualte the (i) radius in cm and (ii) volume in cm³ in terms of π.
Answers
Answer:
(i) 5 cm.
(ii) 100π cm³.
Step-by-step explanation:
Slant height of the cone (l) = 13 cm
(i) Let the radius of the cone be r cm. Then its total surface area:
=> (πrl + πr²)
=> πr(l + r)
=> πr(13 + r) cm²
But, total surface area = 90 π cm² (Given)
∴ πr(13 + r) = 90π
=> r(13 + r) = 90
=> r² + 13r - 90 = 0
=> r² + 18r - 5r - 90 = 0
=> r(r + 18) - 5(r + 18) = 0
=> (r + 18) (r - 5) = 0
∴ r = -18 & 5 [r ≠ -18]
Hence, the radius of the cone is 5 cm.
(ii) Let the height of the cone be h cm. Then,
=> h² = (l² - r²) = (13² - 5²) = 144
=> h = √144 = 12 cm
∴ Volume of cone = 1/3πr²h
=> 1/3π × 5 × 5 × 12
∴ 100π cm³
Hence, the volume of the cone is 100π cm³.
Required Answer :
- The radius of cone = 5 cm
- The volume of cone in terms of π = 100π cm³
Given :
- Total surface area of a right circular cone = 90 π cm²
- Slant height of the cone = 13 cm
To find :
- (i) radius in cm
- (ii) volume in cm³ in terms of π.
Solution :
⇒ Total surface area = Curved surface of cone + area of base
⇒ TSA = πrl + πr²
⇒ 90π = πr(l + r)
⇒ 90 = r(l + r)
⇒ 90 = r(13 + r)
⇒ 90 = 13r + r²
⇒ r² + 13r - 90 = 0
⇒ Product = - 90r²
⇒ r² + 18r - 5r - 90 = 0
⇒ r(r + 18) - 5(r + 18) = 0
⇒ (r - 5)(r + 18) = 0
⇒ r = 5 or r = - 18 Reject - ve
⇒ r = 5
⇒ The value of r = 5
Therefore,
- The radius of cone = 5 cm
Using formula,
- l² = r² + h²
⇒ (13)² = (5)² + h²
⇒ (13 - 5)(13 + 5) = h²
⇒ (8)(18) = h²
⇒ √(8 × 18) = h
⇒ √(2 × 2 × 2 × 2 × 3 × 3) = h
⇒ ± 12 Reject - ve = h
Height of the cone = 12 cm
Volume of the cone :
Using formula,
- Volume of cone = ⅓ πr²h
⇒ Volume = ⅓ × π × 5 × 5 × 12
⇒ Volume = π × 5 × 5 × 4
⇒ Volume = 100π
Therefore,
- The volume of cone in terms of π = 100π cm³