Question

a toy is in the form of a cone of radius 3.5 CM mounted on a hemisphere of the same radius the total height of the toy is 15.5 cm find the total surface area of the toy

Answer 1

Given:

• Radius of the Hemisphere, r = 3.5 cm².
• Height of the toy = 15.5 cm

To find:

• The total surface area of the toy.

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$\setlength{\unitlength}{1cm}\begin{picture}(6, 4)\linethickness{0.26mm}\qbezier(5.8,2.0)(5.8,2.3728)(4.9799,2.6364)\qbezier(4.9799,2.6364)(4.1598,2.9)(3.0,2.9)\qbezier(3.0,2.9)(1.8402,2.9)(1.0201,2.6364)\qbezier(1.0201,2.6364)(0.2,2.3728)(0.2,2.0)\qbezier(0.2,2.0)(0.2,1.6272)(1.0201,1.3636)\qbezier(1.0201,1.3636)(1.8402,1.1)(3.0,1.1)\qbezier(3.0,1.1)(4.1598,1.1)(4.9799,1.3636)\qbezier(4.9799,1.3636)(5.8,1.6272)(5.8,2.0)\put(0.2,2){\line(1,0){5.6}}\put(3,2){\line(0,2){4.5}}\put(1.5,1.7){\sf{3.5 cm}}\qbezier(.2,2.05)(.7,3)(3,6.5)\qbezier(5.8,2.05)(5.3,3)(3,6.5)\put(2,4){\sf 12 cm}\put(3,2.02){\circle*{0.15}}\put(2.7,2){\dashbox{0.01}(.3,.3)}\qbezier(0.2,2)(2.9,-2)(5.8,2)\end{picture}$

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Solution: Finding surface area of the Hemisphere.

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$\star\;\boxed{\sf{\pink{Surface\; area_{\;(Hemisphere)} = 2\pi r^2}}}$

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$:\implies\sf 2 \times \dfrac{22}{7} \times 3.5 \times 3.5 \\\\\\:\implies\sf \dfrac{2 \times 22 \times 35 \times 35}{7 \times 10 \times 10} \\\\\\:\implies\boxed{\sf{77\;cm^2}}$

To calculate Slant Height l,

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• Height of conical part, 15. 5 – 3.5 = 12 cm.

$:\implies\boxed{\sf{\pink{(l)^2 = \sqrt{(r)^2 + (h)^2}}}}$

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where,

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• l is slant Height, r is radius and h is height.

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Therefore,

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$:\implies\sf l^2 = \sqrt{(3.5)^2 + (12)^2} \\\\\\:\implies\sf l^2 = \sqrt{12.25 + 144}\\\\\\:\implies\sf l^2 = \sqrt{156.25} \\\\\\:\implies{\underline{\boxed{\frak{\pink{l = 12.5\;cm}}}}}$

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$\therefore{\underline{\sf{Hence,\;slant\; Height\; is\; \bf{ 12.5\;cm}.}}}$

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$\underline{\bf{\dag} \:\mathfrak{As\;we\;know\: that\: :}}$⠀

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$\star\;\boxed{\sf{\pink{CSA_{\;(cone)} = \pi r l}}}$

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Therefore,

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$:\implies\sf \bigg(\dfrac{22}{7} \times 3.5 \times 12.5 \bigg)\\\\\\:\implies\sf \bigg(\dfrac{22}{7} \times \dfrac{35}{10} \times \dfrac{125}{10} \bigg) \\\\\\:\implies\sf \dfrac{275}{2}\;cm^2$

• Total surface area of the toy = Surface area of Hemisphere + Surface area of the conical part.

Therefore,

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$:\implies\sf 77 + \dfrac{275}{2}\\\\\\:\implies\sf \dfrac{ 154 + 275}{2}\\\\\\:\implies\sf \dfrac{429}{2}\\\\\\:\implies{\underline{\boxed{\frak{\pink{214.5\;cm^2}}}}}\;\bigstar$

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$\therefore{\underline{\sf{Hence, \;TSA\: of \: the \; Toy \; is \; \bf{214.5\;cm^2 }.}}}$

Answer 2

Answer:

Explanation:

Given :

• Radius of cone = 3.5 cm
• Radius of hemisphere = Radius of cone
• Total height of toy = 15.5 cm

To Find :

• The total surface area of the toy.

Solution :

We know,

• CSA of hemisphere = 2πr²

=> CSA = 2 × 3.14 × (3.5)²

=> CSA = 6.28 × 12.25

=> CSA = 77 cm² (Approx)

Now,

• Height of cone = Height of toy - Radius of hemisphere

=> h = 15.5 - 3.5

=> h = 12 cm

We need to find slant height of cone,

• l² = h² + r²

=> l² = 12² + 3.5²

=> l² = 144 + 12.25

=> l² = 156.25

=> l = 12.5 cm

We know that,

• CSA of cone = πrl

=> CSA = 3.14 × 3.5 × 12.5

=> CSA = 3.14 × 43.75

=> CSA = 137.375 cm²

So,

• TSA of toy = CSA of hemisphere + CSA of cone

=> TSA = 77 + 137.375

=> TSA = 214.375 cm²

Hence :

The total surface area of toy is 214.375 cm².