Question

A cylinder and a cone have equal radii of their bases and equal heights. If their curved surface area are in the ratio 8:5, show that the radius of each is to the height of each as 3:4.​

Answer 1

Given :-

• Cylinder and a cone have equal radii of their bases and equal heights.
• CSA ratio = 8 : 5 .

To Show :-

• Radius : Height = 3 : 4 . ?

Formula used :-

• CSA of cone = π * r * l
• l = Slant Height = √(r² + h²) => l² = (r² + h²)
• CSA of cylinder = 2 * π * r * h

Solution :-

→ Let radius of cone & cylinder = r

→ Height of cone & cylinder = h .

→ slant height of cone = l

A/q,

→ (CSA of cylinder) / (CSA of cone ) = 8/5

→ (2 * π * r * h ) / ( π * r * l ) = 8/5

π & r will be cancel,

→ (2h / l) = 8/5

→ h / l = 4/5

Squaring both sides ,

→ (h/l)² = (4/5)²

→ h²/ l² = 16/25

Putting l² = (r² + h²) in LHS,

→ h²/(r² + h²) = 16/25

Cross - Multiply,

→ 25h² = 16(r² + h²)

→ 25h² = 16r² + 16h²

→ 25h² - 16h² = 16r²

→ 9h² = 16r²

→ r²/h² = 9/16

Square - root both sides ,

→ r/h = 3/4.

Hence,

→ r : h = 3 : 4. (Ans).

Answer 2

Given:

A cylinder and a cone have equal radii of their bases and equal heights. their curved surface area are in the ratio 8:5.

To show:

Show that the radius of each is to the height of each as 3:4.

Formula used:

• C.S.A of cylinder = 2πrh
• C.S.A of cone = πrl

Solution:

Let Radius be x and height be h.

C.S.A of cylinder/ C.S.A of cone = 8/5 [Given]

⇒ 2πrh/πrl = 8/5

⇒ 2h/l = 8/5

⇒ 2h × 5 = 8l

⇒ 10h = 8l

⇒ 5h = 4l

We know that,

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

$5h = 4 \sqrt{h {}^{2} + r {}^{2} }$

$\star \: squaring \: both \: sides$

$(5h) {}^{2} = (4 \sqrt{h {}^{2} + r {}^{2} } ) {}^{2}$

$\rightarrow \: 25h {}^{2} = 16 {h}^{2} + 16 {r}^{2}$

$\rightarrow \: 25 {h}^{2} - 16 {h}^{2} = 16 {r}^{2}$

$\rightarrow \: 9 {h}^{2} = 16 {r}^{2}$

$\rightarrow \: \frac{ {r}^{2} }{ {h}^{2} } = \frac{9}{16}$

$\rightarrow \: \frac{r}{h} = \sqrt{ \frac{9}{16} }$

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

$\therefore \: ratio \: of \: their = 3 : \: 4$