A right triangle, whose base and height are 15 cm. and 20 cm. respectively is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed (Use π=3.14).
Answers
When the right angle triangle is revolved about the hypotenuse AC, the figure so formed is a double cone.
Now, from the figure,
In triangle ABC,
BC2 = AB2 + AC2
=> BC2 = (15)2 + (20)2
=> BC2 = 225 + 400
=> BC2 = 625
=> BC = √625
=> BC = 25
Now, let OB = x
then OC = BC - OB
=> OC = 25 - x
Now, in triangle AOB,
AB2 = OA2 + OB2
=> (15)2 = OA2 + x2
=> 225 = OA2 + x2
=> OA2 = 225 - x2 .................1
Again, in triangle AOC,
AC2 = OA2 + OC2
=> (20)2 = OA2 + (25 - x)2
=> 400 = OA2 + (25 - x)2
=> OA2 = 400 - (25 - x)2.................2
Now, from equation 1 and 2, we get
225 - x2 = 400 - (25 - x)2
=> 225 - x2 = 400 - (625 + x2 - 50x)
=> 225 - x2 = 400 - 625 - x2 + 50x
=> 225 = 400 - 625 + 50x
=> 225 = -225 + 50x
=> 225 + 225 = 50x
=> 50x = 450
=> x = 450/50
=> x = 9
=> OB = 9
and OC = 25 - 9 = 16
From equation 1, we get
OA2 = 225 - 92
=> OA2 = 225 - 81
=> OA2 = 144
=> OA = √144
=> OA = 12
Now, volume of the double cone = volume of the smaller cone + volume of the larger cone
=> volume of the double cone = (1/3)×π×(OA)2 ×(OB) + (1/3) ×π×(OA)2 ×(OC)
=> volume of the double cone = (1/3)×π×(OA)2 ×(OB + OC)
=> volume of the double cone =(1/3)×(22/7)×(12)2 × BC
=> volume of the double cone = (1/3)×(22/7)×12×12×25
=> volume of the double cone = (22/7)×4×12×25
=> volume of the double cone = 26400/7 cm3
Again , surface area of the double cone = curved surface area of the smaller cone+curved surface area of the larger cone
=> surface area of the double cone = π ×OA
×AB+ π ×(OA×AC)
=> surface area of the double cone = π×OA×(AB + AC)
=> surface area of the double cone = (22/7)×12×(15 + 20)
=> surface area of the double cone = (22/7)×12×35
=> surface area of the double cone = 22×12×35
=> surface area of the double cone = 1320 cm2
HOPE IT HELPS YOU
Total surface area = 1318.8 cm² cm² & Total Volume = 3768 cm³ if A right triangle, whose base and height are 15 cm. and 20 cm. respectively is made to revolve about its hypotenuse.
Step-by-step explanation:
Right angle triangle is revolved about the hypotenuse BC, the figure so formed is a double cone.
Attached is figure
In triangle ABC,
AB = 15 cm
AC = 20 cm
BC = 25 cm (as BC² = AB² + AC²)
OB = h₁
OC = h₂
OA = OD = r
h₁ + h₂ = 25
15² =h₁² + r²
20² = h₂² + r²
=> 20² - 15² = h₂² - h₁²
=> 400 - 225 = (h₂+ h₁)(h₂ - h₁)
=> 175 = 25(h₂ - h₁)
=> h₂ - h₁ =7
h₁ + h₂ = 25
=> h₁ = 9
=> h₂ = 16
15² =h₁² + r² or 20² = h₂² + r²
=> r = 12
total surface area = Curved surface area of both
= π * 12 * 15 + π * 12 * 20
= π * 420
= 3.14 * 420
= 1318.8 cm²
Total Volume = Volume of both cone
= (1/3)π (12)² * (9) + (1/3)π (12)² * (16)
= ( (1/3)π 3600
= 1200 π
= 1200 * 3.14
= 3768 cm³
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