Question

The base area of a right circular cone is
346.5 cm² and its height is 13 cm. What would be
the total surface area if the radius is halved and the
height is doubled? What is the volume of the new
cone formed?​

Answer 1

Question :

The base area of a right circular cone is 346.5 cm² and its height is 13 cm. What would be the total surface area if the radius is halved and the

height is doubled? What is the volume of the new cone formed?

Answer :

$\bf\pink {Given :}$

• base area of a right circular cone is 346.5 cm Sq.

• height of the cone is 13 cm .

$\bf\pink {To\:find :}$

• Total surface area when radius is halved and height is doubled ?

Explanation :

Base area means the area of the circular base .

$\sf\red {Area\:of\:base = \pi {r}^{2}}$

$\\ \longrightarrow\sf{346.5 = \dfrac {22}{7}\times {r}^{2}}$

$\\ \longrightarrow\sf{346.5\times \dfrac {7}{22} = {r}^{2}}$

$\\ \longrightarrow\sf{15.75\times 7= {r}^{2}}$

$\\ \longrightarrow\sf{110.25 = {r}^{2}}$

$\\ \longrightarrow\sf{r = \sqrt {110.25}}$

$\\ \longrightarrow\sf{r = \sqrt {10.5}}$

• Therefore the radius is 10.5 cm .

let R be the halved radius .

$\sf\purple {R = \dfrac{1}{2}r}$

$\\ \longrightarrow\sf{R = \dfrac{10.5}{2}}$

$\\ \longrightarrow\sf{R = 5.25\:cm}$

• Therefore the halved radius R is 5.25 cm .

let H be the doubled height .

$\sf\purple {H = 2\times h}$

$\\ \longrightarrow\sf{H = 2\times 13}$

$\\ \longrightarrow\sf{H = 26\:cm}$

• Therefore the doubled height is 26 cm .

$\sf\red{Total\:surface\:area\:of\:cone = \pi R ( l + R)}$

• l is the slanting height

$\\ \longrightarrow\sf{l = \sqrt {{R}^{2} + {H}^{2}}}$

$\\ \longrightarrow\sf{l = \sqrt {{(5.25 + 26})^{2}}}$

$\\ \longrightarrow\sf{l = \sqrt {(31.25)^{2}}}$

$\\ \longrightarrow\sf{l = 31.25}$

$\\ \longrightarrow\sf{\dfrac {22}{7}\times 5.25\times (31.25 + 5.25)}$

$\\ \longrightarrow\sf{\dfrac {22}{7}\times 5.25\times 36.5}$

$\\ \longrightarrow\sf{\dfrac {22}{7}\times 191.625}$

$\\ \longrightarrow\sf{22\times 27.375}$

$\\ \longrightarrow\sf{602.25}$

• Therefore the total surface area is 602.25 cm Sq

$\sf\red{volume\:of\:cone = \dfrac {1}{3}\pi {R}^{2}H}$

$\\ \longrightarrow\sf{\dfrac {1}{3}\times \dfrac {22}{7}\times {5.25}^{2}\times 26}$

$\\ \longrightarrow\sf{\dfrac {1}{3}\times \dfrac {22}{7}\times 27.5625\times 26}$

$\\ \longrightarrow\sf{\dfrac {1}{3}\times 22\times 3.9375\times 26}$

$\\ \longrightarrow\sf{ 22\times 1.3125\times 26}$

$\\ \longrightarrow\sf{ 28.875\times 26}$

$\\ \longrightarrow\sf{ 750.75}$

• Therefore the volume is 750.75 cm cube