Answer:
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 [tex]\frac{8\sqrt{3}\pi}{3}[/tex]units³
Step-by-step 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 = [tex]\frac{1}{3}[/tex] (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 = [tex]\frac{1}{3}[/tex] (π r² h)
∵ r = 2 and h = 2√3
∴ V = [tex]\frac{1}{3}[/tex] (π × 2² × 2√3) = [tex]\frac{1}{3}[/tex] (π × 4 × 2√3) = [tex]\frac{1}{3}[/tex] (π × 8√3)
∴ V = [tex]\frac{8\sqrt{3}\pi}{3}[/tex]
* The volume of the cone is [tex]\frac{8\sqrt{3}\pi}{3}[/tex]units³
