Question

The total surface area of a right circular cone of slant height 13 cm is 90 π cm². Calcualte the (i) radius in cm and (ii) volume in cm³ in terms of π.

Answer 1

Answer:

(i) 5 cm.

(ii) 100π cm³.

Step-by-step explanation:

Slant height of the cone (l) = 13 cm

(i) Let the radius of the cone be r cm. Then its total surface area:

=> (πrl + πr²)

=> πr(l + r)

=> πr(13 + r) cm²

But, total surface area = 90 π cm² (Given)

∴ πr(13 + r) = 90π

=> r(13 + r) = 90

=> r² + 13r - 90 = 0

=> r² + 18r - 5r - 90 = 0

=> r(r + 18) - 5(r + 18) = 0

=> (r + 18) (r - 5) = 0

∴ r = -18 & 5 [r ≠ -18]

Hence, the radius of the cone is 5 cm.

(ii) Let the height of the cone be h cm. Then,

=> h² = (l² - r²) = (13² - 5²) = 144

=> h = √144 = 12 cm

∴ Volume of cone = 1/3πr²h

=> 1/3π × 5 × 5 × 12

∴ 100π cm³

Hence, the volume of the cone is 100π cm³.

Answer 2

$\red{\large\underline{\sf{Given- }}}$

The total surface area of a right circular cone is 90 π cm²

Slant height of cone is 13 cm.

$\blue{\large\underline{\sf{To\:Find - }}}$

(i) radius in cm

(ii) volume in cm³ in terms of π.

$\green{\large\underline{\sf{Solution-}}}$

Given that,

The total surface area of a right circular cone of slant height 13 cm is 90 π cm².

So, Let suppose that, the radius of cone be r cm.

We know

$\red{ \boxed{ \sf{ \:TSA_{cone} \: = \: \pi \: r \: (l \: + \: r) \: }}}$

where,

l is the slant height of cone.

r is the radius of cone.

Thus,

Slant height, l = 13 cm

Total Surface Area, TSA = 90 π cm²

So, on Substituting the values, we get

$\rm :\longmapsto\: 90 \: \pi \: = \: \pi \: r(13 + r)$

$\rm :\longmapsto\: 90 = r(13 + r)$

$\rm :\longmapsto\: {r}^{2} + 13r = 90$

$\rm :\longmapsto\: {r}^{2} + 13r - 90 = 0$

$\rm :\longmapsto\: {r}^{2} + 18r - 5r - 90 = 0$

$\rm :\longmapsto\:r(r + 18) - 5(r + 18) = 0$

$\rm :\longmapsto\:(r + 18)(r - 5) = 0$

$\bf\implies \:r \: = \: 5 \: cm \: as \: r \: \ne \: - 18$

Let evaluate the height of cone.

We have now,

Slant height of cone, l = 13 cm

Radius of cone, r = 5 cm

So, Let assume that height of cone be h cm

So, we know that,

$\rm :\longmapsto\: {l}^{2} = {r}^{2} + {h}^{2}$

$\rm :\longmapsto\: {13}^{2} = {5}^{2} + {h}^{2}$

$\rm :\longmapsto\: 169 = 25 + {h}^{2}$

$\rm :\longmapsto\: 169 - 25 = {h}^{2}$

$\rm :\longmapsto\: 144 = {h}^{2}$

$\bf\implies \:h \: = \: 12 \: cm$

Now, we know that,

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

So, on substituting the values of h and r, we get

$\rm :\longmapsto\:Volume_{cone} = \dfrac{\pi}{3} \times {5}^{2} \times 12$

$\bf\implies \:Volume_{cone} = 100\pi \: {cm}^{3}$

Additional Information :-

Volume of cylinder = πr²h

T.S.A of cylinder = 2πrh + 2πr²

Volume of cone = ⅓ πr²h

C.S.A of cone = πrl

T.S.A of cone = πrl + πr²

Volume of cuboid = l × b × h

C.S.A of cuboid = 2(l + b)h

T.S.A of cuboid = 2(lb + bh + lh)

C.S.A of cube = 4a²

T.S.A of cube = 6a²

Volume of cube = a³

Volume of sphere = 4/3πr³

Surface area of sphere = 4πr²

Volume of hemisphere = ⅔ πr³

C.S.A of hemisphere = 2πr²

T.S.A of hemisphere = 3πr²