Question

A wooden toy is in the form of a cone surmounted on a hemisphere. The diameter of the base of the cone is 16 cm and its height is 15 cm. Find the cost of painting the toy at Rs. 7 per 100 cm².

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Answer 1

Given:-

• The diameter of the base of the cone is 16 cm and its height is 15 cm.

Solutions:-

• Find the cost of painting the toy at Rs. 7 per 100 cm².

Solutions:-

• Length of the cone = 15cm
• Diameter of the cone base and hemisphere = 16cm.

Therefore,

Radius of the cone base and hemisphere = 16/2 = 8cm

The slant height of the cone l = √ r²+ h²

=> l = √r² + h²

=> √8² + 15²

=> √64 + 225

=> √289

=> 17cm

Now,

Total surface area = lateral surface area of the cone + surface area of the hemisphere surface area

=> πrl + 2πr²

=> 22/7 × 8 × 17 + 2 × 22/7 × 8²

=> 2992/7 + 2816/7

=> 427.42 + 402.28

=> 829.71cm²

So,

The total surface area of the toy is 829.71cm²

Now,

The rate of painting the toy = Rs 7 per 100cm²

=> Rs 7/100cm²

So,

The cost of painting the = total surface area × rate of painting

=> 829.17 × 7/100

=> 58.08

Hence, the cost of painting the toy is Rs 58.08.

Answer 2

Given :

• A wooden toy is in the form of a cone surmounted on a hemisphere
• Diameter of base of cone = 16 cm ; therefore, Radius of base of cone, r = 8 cm = Radius of Hemisphere
• Height of cone, h = 15 cm
• Rate of painting the toy = 7 per 100 cm²

To find :

• Cost of Painting the toy = ?

Formulae required :

• Formula for CSA of cone

$\red{\bigstar}\boxed{\sf{CSA\:of\:cone=\pi\;r\;l}}$

[ where l is slant height and r is radius of cone ]

• Formula for CSA of hemisphere

$\red{\bigstar}\boxed{\sf{CSA\:of\:hemisphere=2\pi r^2}}$

[ where r is radius of hemisphere ]

Solution :

Calculating slant height of cone

$\implies\sf{l^2=h^2+r^2}$

[ where l is slant height, h is height and r is radius of cone ]

$\implies\sf{l^2=(15)^2+(8)^2}$

$\implies\sf{l^2=225+64}$

$\implies\sf{l^2=289}$

$\implies\underline{\underline{\red{\sf{l=17\;cm}}}}$

Calculating CSA of cone surmounted

$\implies\sf{CSA\:of\:cone=\pi\;r\;l}$

$\implies\sf{CSA\:of\:cone=\pi\times 8 \times 17}$

$\implies\underline{\underline{\red{\sf{CSA\:of\:cone=136\;\pi\;\;cm^2}}}}$

Calculating CSA of hemisphere

$\implies\sf{CSA\:of\:hemisphere=2\;\pi\;r^2}$

$\implies\sf{CSA\:of\:hemisphere=2\times\pi\times8^2}$

$\implies\sf{CSA\:of\:hemisphere=2\times\pi\times64}$

$\implies\underline{\underline{\red{\sf{CSA\:of\:hemisphere=128\;\pi\;\;cm^2}}}}$

Calculating TSA of toy

$\implies\sf{TSA\:of\:toy=CSA\:of\:cone+CSA\:of\:hemisphere}$

$\implies\sf{TSA\:of\:toy=136\;\pi+128\;\pi}$

$\implies\underline{\underline{\red{\sf{TSA\:of\:toy=264\;\pi\;\;cm^2}}}}$

Calculating cost of painting the toy

$\implies\sf{Cost\:of\:painting=Rate\:of\:painting\times TSA\:of\:toy}$

$\implies\sf{Cost\:of\:painting=\dfrac{Rs\;7}{100\;cm^2}\times 264\;\pi\;\;cm^2}$

$\implies\sf{Cost\:of\:painting=\dfrac{Rs\;7}{100}\times 264\;\times \dfrac{22}{7}}$

$\implies\sf{Cost\:of\:painting=\dfrac{Rs\;1}{100}\times 264\;\times 22}$

$\implies\underline{\underline{\boxed{\red{\sf{Cost\:of\:painting=58.08\;\;Rs}}}}}$

Therefore,

• Cost of Painting the Toy is 58.08 Rs.