Question

Find the capacity in litres of a conical vessel with(i) radius7 cm, slant height 25 cm (ii) height 12 cm , slant height 13 cm​

Answer 1

Answer:

Given :-

(i) Radius 7 cm, slant height 25 cm

(ii) Height 12 cm, slant height 13 cm

To Find :-

• What is the capacity in litres of a conical vessel.

Formula Used :-

$\clubsuit$ Volume of Cone Formula :

$\bigstar \: \: \sf\boxed{\bold{\pink{Volume_{(Cone)} =\: \dfrac{1}{3}{\pi}r^2h}}}\: \: \: \bigstar$

where,

• π = Pie or 22/7
• r = Radius
• h = Height

Solution :-

✫ (i) Radius 7 cm, slant height 25 cm :

First, we have to find the height :

Given :

• Radius = 7 cm
• Slant Height = 25 cm

According to the question by using the formula we get,

$\implies \bf (l)^2 =\: (r)^2 + (h)^2$

where,

• l = Slant Height
• r = Radius
• h = Height

By putting the values we get,

$\implies \sf (25)^2 =\: (7)^2 + (h)^2$

$\implies \sf 625 =\: 49 + (h)^2$

$\implies \sf 625 - 49 =\: h^2$

$\implies \sf 576 =\: h^2$

$\implies \sf \sqrt{576} =\: h$

$\implies \sf 24 =\: h$

$\implies \sf\bold{\purple{h =\: 24\: cm}}$

Now, we have to find the capacity of a conical vessel :

Given :

• Radius = 7 cm
• Height = 24 cm

According to the question by using the formula we get,

$\implies \sf Capacity_{(Conical\: Vessel)} =\: \dfrac{1}{3} \times \dfrac{22}{7} \times (7)^2 \times 24$

$\implies \sf Capacity_{(Conical\: Vessel)} =\: \dfrac{22}{21} \times 49 \times 24$

$\implies \sf Capacity_{(Conical\: Vessel)} =\: \dfrac{22}{21} \times 1176$

$\implies \sf Capacity_{(Conical\: Vessel)} =\: \dfrac{25872}{21}$

$\implies \sf\bold{\green{Capacity_{(Conical\: Vessel)} =\: 1232\: cm^3}}$

Now, we have to convert the capacity of a conical vessel into litres :

$\leadsto \sf 1232\: cm^3$

$\leadsto \sf \dfrac{1232}{1000}\: litres$

$\leadsto \sf\bold{\red{1.232\: litres}}$

$\therefore$ The capacity of a conical vessel is 1.232 litres .

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✯ (ii) Height 12 cm, slant height 13 cm :

First, we have to find the radius :

Given :

• Height = 12 cm
• Slant Height = 13 cm

According to the question by using the formula we get,

$\implies \bf (l)^2 =\: (r)^2 + (h)^2$

$\implies \sf (13)^2 =\: (r)^2 + (12)^2$

$\implies \sf 169 =\: r^2 + 144$

$\implies \sf 169 - 144 =\: r^2$

$\implies \sf 25 =\: r^2$

$\implies \sf \sqrt{25} =\: r$

$\implies \sf 5 =\: r$

$\implies \sf\bold{\purple{r =\: 5\: cm}}$

Now, we have to find the capacity of a conical vessel :

Given :

• Radius = 5 cm
• Height = 12 cm

According to the question by using the formula we get,

$\implies \sf Capacity_{(Conical\: Vessel)} =\: \dfrac{1}{3} \times \dfrac{22}{7} \times (5)^2 \times 12$

$\implies \sf Capacity_{(Conical\: Vessel)} =\: \dfrac{22}{21} \times 25 \times 12$

$\implies \sf Capacity_{(Conical\: Vessel)} =\: \dfrac{22}{21} \times 300$

$\implies \sf Capacity_{(Conical\: Vessel)} =\: \dfrac{6600}{21}$

$\implies \sf\bold{\blue{Capacity_{(Conical\: Vessel)} =\: 314.28\: cm^3}}$

Now, we have to convert the capacity of a conical vessel into litres :

$\leadsto \sf 314.28\: cm^3$

$\leadsto \sf \dfrac{314.28}{1000}\: litres$

$\leadsto \sf\bold{\red{0.31428\: litres}}$

$\therefore$ The capacity of a conical vessel is 0.31428 litres .

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