Question

1) 1) If the ratio of the curved surface areas of solid cone and a solid right circular cylinder having same base radii and same height is 5:8, then let us determine the ratio of their base radii and height.​

Answer 1

Let the same base radii be r units and same height be h units

Given :

Ratio of CSAs of cone and cylinder with same height and base = 5:8

. CSA of cone = πrl sq.units

CSA of cylinder = 2πrh sq.units

- πrl / 2πrh = 5/8

→ 1/ 2 h = 5/8

We know that

• |= √(r² +h² )

→ √(r²+ h² )/ 2h = 5/8

Squaring on both sides

- [ √(r² + h²) / 2 h ]² = ( 5 /8 )²

→ (r² + h² ) / 4h² = 25 / 64

→ 64r² + 64h² = 100h²

→ 64r² = 100h² - 64h²

→ 64r² = 36h²

= r²/h² = 36 / 64

→(r/h)² = 36/ 64

Taking square root on both sides

√(r/h)² = √(36 / 64 )

→r/h = 6/8

→ r/h = 3/4

- r:h = 3:4

Therefore the ratio of base radius and height is 3:4.

$\underline \mathsf \green{hope \: it \: helps \: u}$

Answer 2

Answer:

Given :-

• If the ratio of the curved surface areas of solid cone and a solid right circular cylinder having same base radii and same height is 5.8.

Find Out :-

• Ratio of their base radii and height.

Solution :-

Let, the radius of base = r

And, the height = h

Now, slant height (l) = $\sqrt{h² + r²}$

Area of C.S.A of the cone = $πr\sqrt{h² + r²}$

According to the question,

➛ $\dfrac{πr\sqrt{h² + r²}}{2πrh} =\: \dfrac{5}{8}$

➛ $\dfrac{\sqrt{h² + r²}}{2h} =\: \dfrac{5}{8}$

➛ $\dfrac{h² + r²}{4h²} =\: \dfrac{25}{64}$

➛ $1 + \dfrac{r²}{h²} =\: \dfrac{25}{16}$

➛ $\dfrac{r²}{h²} =\: \dfrac{25}{16} - 1$

➛ $\dfrac{r²}{h²} =\: \dfrac{9}{16}$

➡ $\dfrac{r}{h} =\: \dfrac{3}{4}$

➦ Radius of base : Height = 3 : 4

∴ The radius of their base radii and height is 3 : 4.