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Class 10 Class 12

Let the radius and height of the cone be 5k and 12k respectively. Then,
r = 5k
h = 12k

$space space space space space space space space space straight V equals 1 third πt squared straight h equals 314 rightwards double arrow space space space 1 third cross times 3.14 cross times left parenthesis 5 straight k right parenthesis squared cross times left parenthesis 12 straight k right parenthesis equals 314 rightwards double arrow space space space space straight k equals 1 therefore space space space space space straight r equals 5 space cm space space space space space space space space straight h space equals space 12 space cm therefore space space space space space l space equals square root of straight r squared plus straight h squared end root space space space space space space space space space space equals square root of 5 squared plus 12 squared end root space space space space space space space space space space space equals space 13 space cm$

Hence, the slant height and radius of the cone are 13 cm and 5 cm respectively.

129 Views

A semi-circular sheet of metal of diameter 28 cm is bent to form an open conical cup. Find the capacity of the cup.

When a semi-circular sheet is bent to form an open conical cup, the radius of the sheet becomes the slant height of the cup and the circumference of the sheet becomes the circumference of the base of the cone.

therefore space space space space space space space space space space space space l italic space equals 28 over 2 space cm space equals space 14 space cm

Let the base radius of the cup be r cm. Then,
2nr = π x 14

$rightwards double arrow space space space space space space 2.22 over 7. straight r equals 22 over 7.14 rightwards double arrow space space space space space space space space space space space space space space straight r equals 7 space cm$

Let the height of the cup be h cm. Then\

l² + r² + h²

$rightwards double arrow space space space space space space space space space space space space left parenthesis 14 right parenthesis squared equals left parenthesis 7 right parenthesis squared plus straight h squared rightwards double arrow space space space space space space space space space space space space space space straight h equals 7 square root of 3 space cm$

$therefore$ Capacity of the cup

$equals 1 third πr squared straight h equals 1 third.22 over 7. left parenthesis 7 right parenthesis squared.7 square root of 3 equals 1078 over 3 square root of 3 equals 1078 over 3 cross times 1.732 equals 622.36 space cm cubed$

243 Views

Find the volume of the largest right circular cone that can be placed in a cube of edge 7 cm.

For largest cone

Base radius (r) =

7 over 2

cm
Height (h) = 7 cm

therefore

Volume =

1 third πr squared straight h

equals 1 third.22 over 7.7 over 2.7 over 2.7 equals space 539 over 6 space cm cubed

92 Views

A conical vessel whose internal radius is 5 cm and height 63 cm is full of oil. Find the the volume of the oil. If that oil is emptied into a cylindrical vessel with internal radius 10 cm, find the height of the oil in the cylindrical vessel.

For conical vessel
r = 5 cm h = 63 cm
∴ Volume of the oil

$equals 1 third πr squared straight h equals 1 third.22 over 7.5.5.63 equals 1650 space cm cubed$
Let the required height be 4 cm. Then, according to the question, π(10)²H = 1650

$rightwards double arrow space space 22 over 7 cross times 100 straight H equals 1650 rightwards double arrow space space space straight H equals fraction numerator 1650 cross times 7 over denominator 22 cross times 100 end fraction equals 5.25 space cm$