Question

a right circular cylindrical container of base radius 6 cm and height 15 cm is full of ice cream. the ice cream to be filled in cones of height 9 cm and base radius 3 cm, having a hemispherical cap. find the number of cones needed to empty the container​

Answer 1

Answer:

The number of cones needed to empty the container is 12.

Step-by-step-explanation:

We have given the dimensions of a cylindrical container and a cone with hemispherical cap.

For cylindrical container,

Radius ( R ) = 6 cm

Height ( H ) = 15 cm

For cone,

Radius ( r ) = 3 cm

Height ( h ) = 9 cm

Now, we know that,

$\displaystyle{\sf\:No.\:of\:cones\:required\:=\:\dfrac{Volume\:of\:cylindrical\:container} {Volume\:of\:ice-cream\:cone}}$

$\displaystyle{\implies\sf\:No.\:of\:cones\:required\:=\:\dfrac{Volume\:of\:cylindrical\:container}{(\:Volume\:of\:cone\:+\:Volume\:of\:hemisphere\:)}}$

$\displaystyle{\implies\sf\:No.\:of\:cones\:required\:=\:\dfrac{\pi\:R^2\:H}{\left(\:\dfrac{1}{3}\:\pi\:r^2\:h\:\right)\:+\:\left(\:\dfrac{2}{3}\:\pi\:r^3\:\right)}}$

$\displaystyle{\implies\sf\:No.\:of\:cones\:required\:=\:\dfrac{\dfrac{22}{7}\:\times\:6\:\times\:6\:\times\:15}{\left(\:\dfrac{1}{3}\:\times\:\dfrac{22}{7}\:\times\:3\:\times\:3\:\times\:15\:\right)\:+\:\left(\:\dfrac{2}{3}\:\times\:\dfrac{22}{7}\:\times\:3\:\times\:3\:\times\:3\:\right)}}$

$\displaystyle{\implies\sf\:No.\:of\:cones\:required\:=\:\dfrac{\dfrac{22}{7}\:\times\:6\:\times\:6\:\times\:15}{\dfrac{22}{7}\:\times\:3\:\times\:3\:\left(\:\dfrac{1}{\cancel{3}}\:\times\:\cancel{9}\:+\:\dfrac{2}{\cancel{3}}\:\times\:\cancel{3}\:\right)}}$

$\displaystyle{\implies\sf\:No.\:of\:cones\:required\:=\:\dfrac{\cancel{\dfrac{22}{7}}\:\times\:6\:\times\:6\:\times\:15}{\cancel{\dfrac{22}{7}}\:\times\:3\:\times\:3\:(\:3\:+\:2\:)}}$

$\displaystyle{\implies\sf\:No.\:of\:cones\:required\:=\:\dfrac{\cancel{6}\:\times\:6\:\times\:15}{\cancel{3}\:\times\:3\:\times\:5}}$

$\displaystyle{\implies\sf\:No.\:of\:cones\:required\:=\:\dfrac{2\:\times\:\cancel{6}\:\times\:15}{\cancel{3}\:\times\:5}}$

$\displaystyle{\implies\sf\:No.\:of\:cones\:required\:=\:\dfrac{2\:\times\:2\:\times\:\cancel{15}}{\cancel{5}}}$

$\displaystyle{\implies\sf\:No.\:of\:cones\:required\:=\:2\:\times\:2\:\times\:3}$

$\displaystyle{\implies\sf\:No.\:of\:cones\:required\:=\:4\:\times\:3}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:No.\:of\:cones\:required\:=\:12}}}}$

Answer 2

Answer:-

$\small \bf \underline{Given:}$

★★ Radius of the Base of the Cylinder = 6cm

★★ Height of the Cylinder = 15cm

★★ Height of the Cone = 9cm

★★ Radius of Base of Cone = 3cm

$\small \bf \underline{To \: find:}$

We need to find the number of cones

$\small \bf \underline{Formula\:used:}$

» Volume of cylinder = $\bf = π R² H$

» Volume of cone = $\bf = \frac{1}{3} π r² h$

» Volume of hemisphere = $\bf \frac{2}{3} π r³$

where,

R = Radius of the Base of the Cylinder

H = Height of the Cylinder

r = Radius of Base of Cone

h = Height of the Cone

$\small \bf \underline{Solution:}$

Let number of cones be 'n'

» Volume of cylinderical container = n × (Volume of cone + Volume of hemisphere)

$\bf = > πR ²H = n × ( \frac{1}{3} π r² h + \frac{2}{3} π r³)$

$\small\bf = > π R² H = n × \frac{1}{3} × π × r² × (h + 2r)$

$\bf = > 3 R² H = n × r² × (h + 2r)$

$\small\bf = > 3 × 6 × 6 × 15 = n × 3 × 3 × (9 + 6)$

$\bf=> n = 12$

Answer: Number of cones required = 12 cones

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