Question

Rasheed got a playing top (lattu) as his birthday present which surprisingly had no colour on it. he want to colour it with caryons .The top is shaped like a cone surmounted by a hemisphere the entire top is 5 cm in height and the diameter of top is 3.5cm find the area he has to colour? (Take = π=22/7).

Answer 1

Step-by-step explanation:

GiveN :-

• Rasheed want to colour a playing top.
• Structure of the playing top is like a cone surmounted by a hemisphere.
• Total height of the playing top is 5 cm.
• Diameter of the playing top is 3.5 cm.

To FinD :-

• The total area of the playing top that he has to colour.

SolutioN :-

As per the information, we know that the structure of the playing top is like a cone surmounted by a hemisphere. So, we can say that the area to be coloured is equal the sum of CSA of cone and CSA of hemisphere. In simple words, Total area to be coloured = CSA of cone + CSA of hemisphere.

And the unknown values in the question are radius and the slant height of the playing top. So firstly, let us find out these values. Let's do it !

RadiuS :-

• 1/2 = diameter
• 1/2 = 3.5
• 1.75 cm.

The required radius of the playing top is 1.75 cm. Now, let us find out the slant height of the top.

$\rule{300}{2}$

SlanT HeighT :-

The measure of the height of the cone is equal to :

• Total height - Radius
• 5 - 1.75
• 3.25 cm

Now, by putting the values in the formula of slant height of cone we get,

$\sf \dashrightarrow {\sqrt{r^2\ +\ h^2}} \\ \\ \sf \dashrightarrow {\sqrt{(1.75)^2\ +\ (3.25)^2}} \\ \\ \sf \dashrightarrow {\sqrt{3.0625\ +\ 10.5625}} \\ \\ \sf \dashrightarrow {\sqrt{13.625}} \\ \\ \dashrightarrow {\underline{\boxed{\pink{\frak{3.69\ cm}}}}}$

The required slant height of the cone is 3.69 cm. Now, we know the measure of radius and slant height. So, let us find out the CSA of the cone by using the required formula.

$\rule{300}{2}$

$\sf \red \bigstar {\underline{Finding\ CSA\ of\ the\ cone:-}}$

$\sf : \implies {CSA\ of\ cone\ =\ {\pi}rl} \\ \\ \sf : \implies {\dfrac{22}{7} \times 1.75 \times 3.69} \\ \\ \sf : \implies {5.5 \times 3.6} \\ \\ : \implies {\underline{\boxed{\pink{\frak{19.8\ cm^2}}}}}$

Hence, the required CSA of the cone is 19.8 cm². Now, let us find out the CSA of the hemisphere by using the required formula.

$\rule{300}{2}$

$\sf \red \bigstar {\underline{Finding\ CSA\ of\ the\ hemisphere:-}}$

$\sf : \implies {CSA\ of\ hemisphere\ =\ 2{\pi}r^2} \\ \\ \sf : \implies {2 \times \dfrac{22}{7} \times (1.75)^2} \\ \\ \sf : \implies {2 \times \dfrac{22}{7} \times 3.0625} \\ \\ \sf : \implies {2 \times 9.625} \\ \\ : \implies {\underline{\boxed{\pink{\frak{19.25\ cm^2}}}}}$

Hence, the required CSA of the hemisphere is 19.25 cm². So, at last, let us find out the area to be coloured.

$\rule{300}{2}$

$\sf \red \bigstar {\underline{Finding\ area\ to\ be\ coloured:-}}$

$\sf : \implies {CSA\ of\ cone\ +\ CSA\ of\ hemisphere} \\ \\ \sf : \implies {19.8\ +\ 19.25} \\ \\ : \implies {\underbrace{\boxed{\pink{\frak{39.05\ cm^2}}}}_{\scriptsize\blue {\sf{Area\ to\ be\ coloured}}}}$

Hence, the area he has to colour is 39.05 cm².