Question

Find the missing measure for a right circular cone TA is 12 pi and LA is 8 pi

Answer 1

The radius of the base of the cone is 2 units

The slant height of the cone is 4 units

The height of the cone is 2√3 units

The volume of the cone is
`(8√(3)\pi)/(3)`units³

Explanation:

* Lets revise the total surface area and the lateral area of a cone

- The lateral area of cone = π r l , where r is the radius of the base

and l is the slant height of the cone

- The surface area of the cone = π r l + π r², where π r l is the lateral

area and π r² is the base area

- The cone has three dimensions radius (r) , height (h) , slant height (l)

- r , h , l formed right triangle, r , h are its legs and l is its hypotenuse,

then l² = r² + h²

- The volume of the con =
`(1)/(3)` (π r² h)

* Now lets solve the problem

- We will use the total area to find the radius of the base

∵ TA = 12π

∵ TA = LA + πr²

∵ LA = 8π

- Substitute the value of the lateral area in the total area

∴ 12π = 8π + π r² ⇒ subtract 8π from both sides

∴ 12π - 8π = π r²

∴ 4π = π r² ⇒ divide both sides by π

∴ r² = 4 ⇒ take square root for both sides

∴ r = 2

* The radius of the base of the cone is 2 units

- We will use the lateral area to find the slant height

∵ LA = π r l

∵ LA = 8π

∵ r = 2

∴ π (2) l = 8π ⇒ divide both sides by π

∴ 2 l = 8 ⇒ divide both sides by 2

∴ l = 4

* The slant height of the cone is 4 units

- Use the rule l² = r² + h² to find the height of the cone

∵ r = 2 and l = 4

∵ l² = r² + h²

∴ (4)² = (2)² + h²

∴ 16 = 4 + h² ⇒ subtract 4 from both sides

∴ 12 = h² ⇒ take square root for both sides

∴ h = √12 = 2√3

* The height of the cone is 2√3 units

∵ The volume of the con =
`(1)/(3)` (π r² h)

∵ r = 2 and h = 2√3

∴ V =
`(1)/(3)` (π × 2² × 2√3) =
`(1)/(3)` (π × 4 × 2√3) =
`(1)/(3)` (π × 8√3)

∴ V =
`(8√(3)\pi)/(3)`

* The volume of the cone is
`(8√(3)\pi)/(3)`units³