Question

The volume of a right circular cone is 9856cm2. If diameter of the base is 28cm. Find a) Height of the cone. b) Slant height.
c)Curved surface area. d)Total surface area.

Answer 1

$\red{\textbf{ANSWER}}$:-

Step-by-step explanation:

$\underline\blue{\bold{Given- }}$

$\blue{\texttt{volume \: of \: a \: right \: circular \: cone \: is }}$$9856 {cm}^{2}$

$\blue{\texttt{diameter \: of \: the \: base \: is }}$ 28cm .

$so \: \: r = \large\frac{28}{2}$

= 14 cm

$\underline\blue{\bold{To \: Find }}$:-

a) Height of the cone.

b) Slant height.

c)Curved surface area

d)Total surface area.

$\red{\textbf{(i)height}}$ = $\underline\blue{\bold{h}}$

$\red{\textbf{volume}}$ =

$9856 = \large \frac{1}{3} \pi {r}^{2} h$

$9856 = \frac{1}{3} \times \frac{22}{7} \times 14 \times 14 \times h$

$h = 48 \: cm$

$\red{\textbf{(ii) \: slant \: height}}$ =

${l}^{2} = \sqrt{ {r}^{2} + {h}^{2} }$

${l}^{2} = \sqrt{ {14}^{2} + {48}^{2} } = \sqrt{196 + 2304}$

${l }^{2} = 2500$

$l = \sqrt{2500}$

$l = \sqrt{(5 {0)}^{2} }$

$l \: = 50 \: cm$

$\red{\textbf{(iii) \: curved \: surface \: Area}}$ =>

we know that r = 14 cm and l = 50cm

$(\pi \: rl \: = \large \frac{22}{7} \times 14 \times 50) {cm}^{2}$

$= (22 \: \times 2 \times 50)c {m}^{2}$

$= 2200 \: c {m}^{2}$

$\red{\textbf{(iv) \: total \: surface \: area}}$ = πr(l + r)

= 22/7 × 14(50 + 14)

= 22 × 2(64)

= 22 × 2 × 64

$= 8192 \: c {m}^{2}$

•$\blue{\texttt{height \: of \: cone}}$ 14cm

•$\blue{\texttt{slant \: height \:}}$$50 {cm}^{2}$

•$\blue{\texttt{curved \: surface \: area}}$ $2200 {cm}^{2}$

•$\blue{\texttt{total \: surface \: area}}$$8192 \: c {m}^{2}$

Answer 2

Given :

The volume of a right circular cone is 9856 cm³. Diameter of the base is 28 cm.

To FinD :

• Height of the cone
• Slant height.
• Curved surface area
• Total surface area

Solution :

Analysis :

We are given with the volume of the cone and diameter of the cone. So from that we can find the height and then by using respective formulas we can find the Height, Slant height, Curved Surface area, Total surface area.

Required Formula :

• Volume of cone = 1/3πr²h

• Slant Height (l) = √[r² + h²]

• CSA = πrl

• TSA = πr(l + r)

where,

• r = radius
• h = height
• π = 22/7 or 3.14
• l = Slant height

Explanation :

a)) Height of the cone :

Diameter = 28 cm

So,

Radius = Diameter/2

= 28/2

= 14 cm

• Diameter = 14 cm

Let us assume that the height is "h" cm.

We know that if we are given the Volume and radius of the cone that is asked to find the height then our required formula is,

Volume of cone = 1/3πr²h

where,

• π = 22/7
• r = 14 cm
• h = h cm
• Volume = 9856 cm³

Using the required formula and substituting the values,

⇒ Volume of cone = 1/3πr²h

⇒ 9856 = 1/3 × 22/7 × 14 × 14 × h

⇒ 9856 = 1/3 × 22 × 2 × 14 × h

⇒ 9856 = 1/3 × 616 × h

⇒ 9856 = 616/3 × h

⇒ 9856 × 3/308 = h

⇒ 29568/616 = h

⇒ 48 = h

∴ Height = 48 cm.

Verification :

⇒ Volume of cone = 1/3πr²h

⇒ 9856 = 1/3 × 22/7 × 14 × 14 × 48

⇒ 9856 = 1/3 × 22 × 2 × 14 × 48

⇒ 9856 = 1/3 × 29568

⇒ 9856 = 29568/3

⇒ 9856 = 9856

∴ LHS = RHS.

• Hence verified.

__________________________________

b)) Slant Height :

We know that if we are given the Height and radius of the cone that is asked to find the slant height then our required formula is,

Slant Height (l) = √[r² + h²]

where,

• π = 22/7
• r = 14 cm
• h = 48 cm

Using the required formula and substituting the values,

⇒ l = √[r² + h²]

⇒ l = √[(14)² + (48)²]

⇒ l = √[196 + 2304]

⇒ l = √[2500]

⇒ l = 50

∴ Slant Height (l) = 50 cm.

__________________________________

c)) Curved Surface Area :

We know that if we are given the slant height and radius of the cone that is asked to find the curved surface area then our required formula is,

Curved Surface area = πrl

where,

• π = 22/7
• r = 14 cm
• l = 50 cm

Using the required formula and substituting the values,

⇒ CSA = πrl

⇒ CSA = 22/7 × 14 × 97

⇒ CSA = 22 × 2 × 50

⇒ CSA = 2200

∴ Curved Surface area = 2200 cm².

__________________________________

d)) Total Surface Area :

We know that if we are given the slant height and radius of the cone that is asked to find the total surface area then our required formula is,

Total Surface area = πr(l + r)

where,

• π = 22/7
• r = 14 cm
• l = 50 cm

Using the required formula and substituting the values,

⇒ TSA = πr(l + r)

⇒ TSA = 22/7 × 14(50 + 14)

⇒ TSA = 22 × 2(64)

⇒ TSA = 22 × 2 × 64

⇒ TSA = 8192

∴ Total Surface area = 8192 cm².

a)) Height = 48 cm.

b)) Slant Height = 50 cm.

c)) Curved Surface area = 2200 cm².

d)) Total Surface area = 8192 cm².