Question

6. The total surface area of a right cone is 1,760 cmand the radius of its base is 14 cm. Find the
following for the cone.
(a) height
​

Answer 1

Answer:

Height of right circular cone = 21.9 cm

Step-by-step explanation:

Given that:

• The total surface area of a right circular cone is 1,760 cm²
• The radius of its base is 14 cm.

To Find:

• The height of the right circular cone.

Formula used:

T.S.A = πr(r + l) sq. unit

Where,

• T.S.A = Total surface area of right circular cone.
• r = Radius of right circular cone.
• l = Slant height of right circular cone.
• π = 22/7

Finding the slant height of right circular cone:

→ 1760 = 22/7 × 14 × (14 + l)

→ 1760 = 44 × (14 + l)

→ 14 + l = 1760/44

→ 14 + l = 40

→ l = 40 - 14

→ l = 26

∴ Slant height = 26 cm

Finding the height of right circular cone:

By using pythagoras theorem,

→ (Slant height)² = (Radius)² + (Height)²

→ (Height)² = (Slant height)² - (Radius)²

→ (Height)² = (26)² - (14)² cm

→ (Height)² = 676 - 196 cm

→ (Height)² = 480 cm

→ Height = √480 cm

→ Height = 21.9 cm (approx.)

Answer 2

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${\large{\bold{\rm{\underline{Proper \: question}}}}}$

The total surface area of a right cone is 1,760 cm and the radius of its base is 14 cm. Find the height for the cone.

${\large{\bold{\rm{\underline{Given \: that}}}}}$

⟹ The total surface area of a right cone is 1,760 cm

${\sf{:\implies Height \: of \: a \: right \: cone \: = \: 14 \: cm}}$

${\large{\bold{\rm{\underline{To \: find}}}}}$

${\sf{:\implies Height \: of \: a \: right \: cone}}$

${\large{\bold{\rm{\underline{Solution}}}}}$

${\sf{:\implies Height \: of \: a \: right \: cone \: = \: 21.9 \: cm}}$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━

${\large{\bold{\rm{\underline{Using \; concept}}}}}$

${\sf{:\implies Formula \: to \: find \: TSA \: of \: cone}}$

${\sf{:\implies Pythagoras \: Theorm}}$

${\large{\bold{\rm{\underline{Using \; formula}}}}}$

${\sf{:\implies TSA \: of \: cone \: = \pi r(r+l)}}$

${\sf{:\implies Height^{2} \: = Slant \: height^{2} \: - Radius^{2}}}$

${\large{\bold{\rm{\underline{Where,}}}}}$

${\sf{:\implies TSA \: denotes \: Total \: Surface \: Area}}$

${\sf{:\implies \pi \: is \: pronounced \: as \: pi}}$

${\sf{:\implies Value \: of \: \pi \: is \: 22/7 \: or \: 3.14}}$

${\sf{:\implies r \: denotes \: radius}}$

${\sf{:\implies l \: denotes \: slant \: height}}$

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${\large{\bold{\rm{\underline{Full \; Solution}}}}}$

~ By using the formula to find TSA of cone, let's find the slant height..!

${\sf{:\implies TSA \: = \pi r(r+l)}}$

${\sf{:\implies 1760 \: = 3.14(14)(14+l)}}$

${\sf{:\implies 1760 \: = 3.14 \times 14 \times (14+l)}}$

${\sf{:\implies 1760 \: = 43.96 \times (14+l)}}$

${\sf{:\implies 1760/43.96 \: = (14+l)}}$

${\sf{:\implies 40 \: = (14+l)}}$

${\sf{:\implies 40 \: = 14+l}}$

${\sf{:\implies 40 - 14 \: = l}}$

${\sf{:\implies 26 = \: l}}$

${\sf{:\implies l \: = 26 \: cm}}$

${\small{\boxed{\boxed{\bf{Slant \: height \: is \: 26 \: cm}}}}}$

~ Now by using phythagoras theorm let's find the height of the cone..!

${\sf{:\implies Height^{2} \: = Slant \: height^{2} \: - Radius^{2}}}$

${\sf{:\implies Height^{2} \: = 26^{2} - 14^{2}}}$

${\sf{:\implies Height^{2} \: = 676 - 196}}$

${\sf{:\implies Height^{2} \: = 480}}$

${\sf{:\implies Height \: = \sqrt{480}}}$

${\sf{:\implies Height \: = 21.9 \: cm}}$

${\small{\boxed{\boxed{\bf{Height \: is \: 21.9 \: cm}}}}}$

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