Question

A toy is in the form of a cone of radius 3.5 cm mounted on a hemispere of same radius . The total height of the toy is 15.5 cm find the total surface area of the toy.

Answer 1

• The T.S.A. (total surface area) of the toy = 214.5 cm².

Given :

A toy is in the form of a cone mounted on a hemisphere.

• Radius of the cone $\bf{\bigg(r_{\sf(cone)}\bigg)}$ = Radius of hemisphere $\bf{\bigg(r_{\sf(hemisphere)}\bigg)}$ = 3.5 cm.
• The total height of the toy (H) = 15.5 cm.

To Find :

• The total surface area (T.S.A.) of the toy.

Solution :

We have to find the total surface area (T.S.A.) of the toy.

Given,

Radius of cone $\bf{\bigg(r_{\sf(cone)}\bigg)}$ = Radius of hemisphere $\bf{\bigg(r_{\sf(hemisphere)}\bigg)}$ = 3.5 cm.

Total height of the toy which is in the form of a cone mounted on a hemisphere (H) = 15.5 cm.

$\star \: \: \: \: \boxed{ \bf{T.S.A. \: of \: toy = C.S.A. \: of \: Hemisphere + L.S.A. \: of \: Cone}}$

We know that,

• C.S.A. of Hemisphere (Curved surface area) = 2πr².
• L.S.A. of Cone (Lateral surface area) = πrl.

First, we need to find the height of the cone.

$\star \: \: \: \: \boxed{\bf{h = H - r_{ \sf{(hemisphere)}}}}$

Where,

• h = height of the cone.
• H = Total height of the toy = 15.5 cm.
• $\bf{\bigg(r_{\sf{(hemisphere)}}\bigg)}$ = radius of hemisphere = 3.5 cm.

Now, substitute the values in the formula of height of the cone.

$\bf \implies h = 15.5 \: cm - 3.5 \: cm \\ \\ \\ \bf \implies h = 12 \: cm$

Therefore, the height of the cone is 12 cm.

Now, we need to find the slant height of the cone.

We know that,

$\star \: \: \: \: \boxed{ \bf{l = \sqrt{ {r}^{2} + {h}^{2} } }}$

Where,

• l = slant height of the cone.
• r = radius of the cone = 3.5 cm.
• h = height of the cone = 12 cm.

Now, substitute both the values in the formula of slant height of the cone.

$\bf \implies l = \sqrt{{(3.5 \: cm)}^{2} + {(12 \: cm)}^{2} } \\ \\ \\ \bf \implies l = \sqrt{12.25 \: {cm}^{2} + 144 \: {cm}^{2}} \\ \\ \\ \bf \implies l = \sqrt{156.25 \: {cm}^{2} } \\ \\ \\ \bf \implies l = 12.5 \: cm$

Therefore, the slant height (l) is 12.5 cm.

Now we have,

• r = radius of hemisphere and cone = 3.5 cm.
• l = slant height of the cone = 12.5 cm.
• The value of π = 22 / 7.

Now, substitute all the values in the formula of T.S.A. (total surface area) of the toy.

$\bf \implies 2\pi {r}^{2} + \pi rl \\\\\\\bf \implies \pi r (2r + l) \\\\\\\bf \implies \frac{22}{7} \times 3.5 \: cm \: (2 \times 3.5 \: cm + 12.5 \: cm) \\\\\\\bf \implies \dfrac{22}{\not7} \times \dfrac{\not35}{10} \: cm\: (7 \: cm + 12.5 \: cm) \\\\\\\bf \implies 22\times\dfrac{\not5}{\not10} \: cm \: (19.5 \: cm) \\\\\\\bf\implies\not22 \times \dfrac{1}{\not2} \: cm \: (19.5 \: cm) \\\\\\\bf \implies 11 \: cm \times 19.5 \: cm \\\\\\\bf \implies 214.5 \: {cm}^{2}$

Hence,

The T.S.A. (total surface area) of the toy is 214.5 cm².

Answer 2

Answer:

Step-by-step explanation:

Total surface area of toy =  Curved surface area of cone +  surface area of hemisphere

Curved  surface area of cone = πrl

Where r=3.5cm, Height =15.5−3.5=12cm

And hence, l=12.5cm       (by using formula l  Sq =h  Sq +b Sq   )

Therefore C.S.A. of cone =π×3.5×12.5

=137.5cm  Sq

 Surface area of hemisphere =2πr  Sq

 =2×π×(3.5)  Sq

 =77cm  Sq

 Hence T.S.A of toy =77+137.5=214.5cm  Sq

Hope it helps!

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