Question

URGENT !!!!


The radius and slant height of a cone are in the ratio 3:5 and its curved surface area is 423.9 cm2. Find the volume of the cone. (π = 3.14)
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Answer 1

Given that,

Ratio of radius and slant height = 3:5

Curved Surface Area = 423.9 cm²

Value of $\pi$ = 3.14

To find,

The volume of the cone = ?

For finding the area of the cone, we must know the value of radius and slant height.

But here, the value is given in the ratio. So, we've to find the value of it.

Assumption:

Let x be the value of radius and slant height.

So, radius becomes = 3x

Slant height becomes = 5x

As we know,

Curved Surface Area = $\pi r l$

Curved Surface Area = $3.14 \times 3x \times 5x$

Curved Surface Area = $3.14 \times 15x^2$

$423.9\:cm^2 = 3.14 \times 15x^2$

$\frac{423.9}{3.14} = 15x^2$

$135 = 15x^2$

$\frac{135}{15} = x^2$

$9 = x^2$

$x^2 = 9$

$x = \sqrt{9}$

$\boxed{\bold{\mathsf{x = 3}}}$

So, the value of x is 3.

Substituting x = 3 in radius and slant height.

Radius = 3x = 3(3) = 9 cm

Slant height = 5x = 5(3) = 15 cm

Therefore, the radius is 9 cm and slant height is 15 cm.

Now we gotta the value of slant height and radius. But for reckoning the volume, we must know the value of height.

But we can reckon by using the following formula,

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

$h = \sqrt{(15)^2 - (9)^2}$

$h = \sqrt{225 - 81}$

$h = \sqrt{144}$

$\boxed{\bold{\mathsf{h = 12\:cm}}}$

So, we've gotta the value of height which 12 cm.

Now we can reckon the volume.

We know that,

Volume of cone = $\pi r^2 \frac{h}{3}$

Volume of cone = $3.14 \times (9)^2 \times \frac{12}{3}$

Volume of cone = $3.14 \times (9)^2 \times \frac{\cancel{12}}{\cancel{3}}$

Volume of cone = $3.14 \times (9)^2 \times 4$

Volume of cone = $3.14 \times 81 \times 4$

Volume of cone = $3.14 \times 324$

$\boxed{\bold{\mathsf{Volume\:of\:cone = 1,017.36\:cm^3}}}$

So, the volume of the cone is 1,017.36 cm³.