Question

the cost of painting the total surface area of a cone at 25 paise per cm² is rupees 176. Find the volume of the cone, if its slant height is 25 cm​

Answer 1

ǫᴜᴇsᴛɪᴏɴ-:

the cost of painting the total surface area of a cone at 25 paise per cm² is rupees 176. Find the volume of the cone, if its slant height is 25 cm

sᴏʟᴜᴛɪᴏɴ-:

T.S.A. = $\sf\frac{176}{0.25}$ = $\sf\frac{176×100}{25}$ = $\sf{70cm^2}$

T.S.A. = $\sf{r~π}$(r + l)

$\sf{70r}$ = $\sf\frac{22}{7}$ × r (r + $\sf{25}$)

$\sf{224}$ = $\sf{r^2}$ + $\sf{25r}$

$\green\implies$$\sf{r^2}$ + $\sf{25r}$ - $\sf{224}$ = $\sf{0}$

$\green\implies$$\sf{r^2}$ + $\sf{32}$ - $\sf{7r}$ - $\sf{224}$ = $\sf{0}$

$\green\implies$$\sf{r(r+32)}$ - $\sf{7(r+32)}$ = $\sf{0}$

$\green\implies$$\sf{(r7)(v+32)}$ = $\sf{0}$

Hence,$\sf{r~=~7cm}$

ᴠᴏʟᴜᴍᴇ-:

V = $\sf\frac{1}{3}$$\sf{πr^2~b}$

= $\sf\frac{1}{3}$$\sf\frac{22}{7}$$\sf{×7×7×24}$

V = $\sf{1232cm^3}$

ʙᴇ ʙʀᴀɪɴʟʏツ

Answer 2

Answer:

Step-by-step explanation:

Total surface area of right circular cone = 704cm²

Slant height (l) = 25cm

Radius of the cone (r) = ?

Total surface area of cone = πrl + πr²

⇢ πrl + πr² = 704

⇢ πr ( l + r ) = 704

⇢ r ( l + r) = $\dfrac{704}{\pi}$

⇢ r ( 25 + r) = $704 \times \dfrac{7}{22}$

⇢ 25r + r² = 224

⇢ r² + 25r - 224 = 0

⇢ r² + 32r - 7r - 224 = 0

⇢ r( r + 32) - 7( r + 32) = 0

⇢ ( r - 7) ( r + 32) = 0

⇢ r - 7 = 0 (or) r + 32 = 0

⇢ r = 7 (or) r = -32

Since, radius cannot be negative

Radius of the cone (r) = 7cm

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

$\rm \dashrightarrow h = \sqrt{ 25^{2} - 7^{2}}$

$\rm \dashrightarrow h = \sqrt{ 625 - 49 }$

$\rm \dashrightarrow h = \sqrt{576}$

$\rm \dashrightarrow h = 24cm$

Now, Volume of the cone = $\rm \dfrac{1}{3} \pi r^{2} h$

= $\rm \dfrac{1}{3} \times \dfrac{22}{7} \times 7^{2} \times 24$

= $\rm \dfrac{22}{7} \ times 7 \times 7 \times 8$

= 22 × 7 × 8

= 1232

$\underline{\boxed{\rm \therefore Volume \: of \: the \: cone = 1232cm^{3}}}$