The sector angle formed by the cone when it is opened is: 54°.
What is the Curved Surface Area of a Cone?
Curved surface area of a cone = πrl, where l is the slant height and r is the radius of the cone.
What is the Area of Sector?
Area of a sector in a circle = ∅/360 × πr², where ∅ is the sector angle.
Step 1: Find the height (h) of the cone using the volume formula, V= 1/3 x π x r² x h:
V = 95.4 cm³
r = 2.4 cm
Plug in the values:
95.4 = 1/3 x 3.14 x 2.4² x h
95.4 = 6.03 x h
h = 15.8 cm
Step 2: Use the pythagroean theorem to find the slant height (l) of the cone
l = √(h² + r²)
Plug in the values
l = √(15.8² + 2.4²)
l = 16 cm.
Step 3: Find the curved surface area of the cone
Curved surface area = πrl = π(2.4)(16) = 120.6 cm².
Step 4: Find the sector angle of the sector formed by the cone when opened
Curved surface area of the cone = area of the sector formed by the cone = 120.6 cm².
Area of a sector in a circle = ∅/360 × πr², therefore:
120.6 = ∅/360 × (3.14)(16²)
120.6 = ∅/360 × 803.84
(120.6)(360) = (∅)(803.84)
43,416 = (∅)(803.84)
43,416/803.84 = ∅
∅ = 54°
Therefore, the sector angle formed by the cone when it is opened is: 54°.
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