Question

find the volume of a cone its total surface area is 7128sq cm and radius of base is 28cm
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Answer 1

Question  find the volume of a cone its total surface area is 7128sq cm and radius of base is 28cm  

TSA = 7128cm^2

Radius = 28cm

TSA = pi r (l+r) = 7128

l+28 = 7128×7/22×28

l+28= 81

l=81-28

l = 53cm

Now ,

h = √l^2-r^2

h = √ 53^2 - 28^2

h = √ 2025

h = 45cm

volume = 1/3 pi r^2 h

volume = 1/3×22/7×28×28×45

volume = 36960cm^3

Answer 2

Given :

• Total surface area of cone = 7128 cm²
• Radius of base = 28 cm

To Find :

• Volume of the cone

Solution :

Let the height of the cone be h cm.

★ Formula :

$\bold{\large{\boxed{\mathtt{\blue{TSA_{cone}\:=\:\pi\:r\:(r\:+\:l)}}}}}$

★ Where,

• r = radius of cone
• l = slant height of cone

$\longrightarrow$$\mathtt{7128\:=\:\pi\:r\:(l+r)}$

$\longrightarrow$ $\mathtt{7128\:=\:28\pi\:(l\:+\:28)}$

$\longrightarrow$ $\mathtt{7128\:=\:28\:\times{\dfrac{22}{7}\:\:(l+28)}}$

$\longrightarrow$ $\mathtt{7128\:=\:4\:\times\:22\:(l+28)}$

$\longrightarrow$ $\mathtt{7128\:=\:88\:(l+28)}$

$\longrightarrow$ $\mathtt{7128\:=\:88l\:+\:2464}$

$\longrightarrow$ $\mathtt{7128-2464\:=\:88l}$

$\longrightarrow$ $\mathtt{4664=88l}$

$\longrightarrow$ $\mathtt{\dfrac{4664}{88}\:=\:l}$

$\longrightarrow$ $\mathtt{53\:=\:l}$

$\bold{\large{\boxed{\rm{\pink{Slant\: height\: of\: cone\: =\: l\: = \:53 \:cm}}}}}$

★ Now, we need to figure out h of the cone to calculate the volume of the cone.

$\longrightarrow$ $\mathtt{l^2\:=\:r^2\:+\:h^2}$

$\longrightarrow$ $\mathtt{53^2\:=\:28^2\:+\:h^2}$

$\longrightarrow$ $\mathtt{2809\:=\:784\:+\:h^2}$

$\longrightarrow$ $\mathtt{2809\:-784\:=\:h^2}$

$\longrightarrow$ $\mathtt{2025\:=\:h^2}$

$\longrightarrow$ $\mathtt{\sqrt{2025}\:=\:h}$

$\longrightarrow$ $\mathtt{45\:=\:h}$

$\bold{\large{\boxed{\mathtt{\green{Height\: of\: cone\: =\: h\: = \:45 \:cm}}}}}$

★ Formula :

$\bold{\large{\boxed{\rm{\red{Volume_{cone}\:=\:{\dfrac{1}{3}\:\:\pi\:r^2\:h}}}}}}$

★ Where,

• r = radius = 28 cm
• h = height = 45 cm

$\longrightarrow$ $\mathtt{\dfrac{1}{3}\:\pi\:\times\:28^2\:\times\:45}$

$\longrightarrow$ $\mathtt{\dfrac{1}{3}\times}$ $\mathtt{\dfrac{22}{7}\:\times}$ $\mathtt{28\:\times\:28\:\times\:45}$

$\longrightarrow$ $\mathtt{\dfrac{776160}{21}}$

$\longrightarrow$ $\mathtt{36960}$

$\bold{\large{\boxed{\rm{\purple{Volume\:of\:cone\:=\:36960\:cm^3}}}}}$