Question

The radius of a come is 75% of its height . If the conical tank occupies 4116000 litres of water , find the curved surface area of the conical tank .

Answer 1

Given :

Vol of water occupied by the conical tank =4116000 l

Radius of cone = 75% of height of the cone

To Find :

Curved surface area of the cone = ?

Solution :

Let the height of the conical tank  be = h

And the radius of the conical be = r

Since radius is 75% of height cone , so :

r = $\frac{75}{100}h$

r =$\frac{3}{4}h$

We know that volume  of cone or conical tank is = $\frac{1}{3}\pi r^2h$

Since we have :

v = 4116000 l

V= 4116 $m^3$

So, $\frac{1}{3}\pi r^2h= 4116 m^3$

Or, $\pi (\frac{3h}{4})^2h = 3 \times 4116$

Or, $\pi h^3 \times 9 = 16 \times 3 \times 4116$

Or, $h^3 =\frac{4116 \times 16 \times 3 }{9 \pi}$

Or, h = 19.12

So, r  = $\frac{3}{4}\times 19.12$

  r = 14.34 m

We know that formula for CSA of a cone is = $\pi r l$

So, we have to find the slant height (l) of the cone :

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

$l = \sqrt{(14.34)^2 + (19.12)^2}$

$l = \sqrt {205.63 + 365.57}$

$l = \sqrt{571.204}$

So, l = 23.9

Now , CSA = $\pi r l$

Or, $CSA = 3.14 \times 14.34 \times 23.9$

So, CSA = 1076 $m^2$

Hence , CSA of cone is $1076 m^2$ .

Answer 2

Given :

The radius of cone = 75% of height of cone

The volume of water occupied = V = 4116000 liters = 4116 × $10^{6}$  cm³

To Find :

The Curved surface area of conical tank

Solution :

Let The height of cone = h cm

The radius of cone = r = 75% of height

                                  r = 0.75 h

∵  Volume of conical tank = V = $\dfrac{1}{3}$ × π × radius² × height

Or,  V = $\dfrac{1}{3}$ × π × r² × h

Or,  4116 × $10^{6}$  cm³ = $\dfrac{1}{3}$ × 3.14 × (0.75 h)² × h

Or, 4116 × $10^{6}$  cm³ = $\dfrac{1}{3}$ × 3.14 × 0.5625  × h³

Or,  4116 × $10^{6}$  cm³ = 0.58875  × h³

∴   h³ = $\dfrac{4116\times 10^{6}}{ 0.58875}$

i.e   h³ = 6991082803

Or,   h = $\sqrt[3]{6991082803}$

Or, Height = h = 1912.11 cm

So , Radius = 0.75 × 1912.11 cm

                  = 1434.08 cm

Again'

∵  Curved surface area of cone = A = π × radius × lateral height

And

Lateral height = l = $\sqrt{radius^{2}+height^{2} }$

                           = $\sqrt{(1434.08)^{2}+(1912.11)^{2} }$

                           = $\sqrt{2056585.4+3656164.65}$

                           = 2390.1  cm

So,   Curved surface area of cone = π × radius × lateral height

                                                         = 3.14 × 1434.08 cm × 2390.1 cm  

                                                         = 10762647.07 cm²

Hence, The Curved surface area of cone is 10762647.07 cm²  Answer