Question

93. A test tube consists of a hemisphere and a cylinder of the same radius. The volume of
water required to fill the whole tube is 2849/3 cm and 2618/3 cm? of water is required
to fill the tube to a level which is 2 cm below the top of the tube. Find the radius of the
tube and the length of its cylindrical part.​

Answer 1

The height is 20 cm and radius is 3.5cm

Given that, the volume of water required to fill the entire tube is = 2849/3 cm³

Also, volume of water filled upto 2cm is = 2618/3 cm³.

Let us consider r and h to be the radius and height of the test tube.

So, volume = (2/3)Пr³ + П²h

This is equal to 2849/3 cm³.

So, (2/3)Пr³ + Пr²h =  2849/3

⇒ Пr²(2r+3h) = 8547/3           ...(1)

As water is filled upto 2 cm, h' = (h-2) cm

(2/3)Пr³ + Пr²(h-2) =  2618/3

⇒ Пr²(2r/3 + h - 2) = 2618/3

⇒ Пr²(2r+3h-6) =  7854/3     ...(2)

Dividing (1) by (2),

Пr²(2r+3h)/Пr²(2r+3h-6) = 8547/7854

⇒ (2r+3h)/(2r+3h-6) = 8547/7854

(1) - (2),

6Пr² =  8547/3 - 7854/3 = 693/3 = 231

r² = 231/6П = 12.25

r = √12.25 = 3.5 cm

Replacing the value of r in (1),

П(3.5)²(2[3.5]+3h) = 8547/3

⇒ 7+3h = 67

⇒ 3h = 67-7 = 60

h = 20cm

Answer 2

The radius of the  tube and the length of its cylindrical part is 2.47 cm and 10.36 cm.

Step-by-step explanation:

The volume of  water filled in the test tube = 2849/3 cm³

The volume of water filled up to 4 cm = 2618/3 cm³

Let radius of the test tube be r

Let height of the test tube be h

2/3 πr³ + πr²h = 2849/3 cm³

πr² (2/3 r + h) = 2849/3 cm³

πr²/3 (2r + 3h) = 2849/3 cm³

⇒ πr² (2r + 3h) = 2849 cm³ → (equation 1)

Now,

2/3 πr³ + πr²(h - 4) = 2618/3 cm³

πr² (2/3 r + h - 4) = 2618/3 cm³

πr²/3 (2r + 3h - 12) = 2618/3 cm³

⇒ πr² (2r + 3h - 12) = 2618 cm³ → (equation 2)

On dividing equation (1) by (2), we get,

(2r + 3h)/(2r + 3h - 12) = 2849/2618 → (equation 3)

On subtracting equation (2) from (1), we get,

πr²(12) = 2849 - 2618 = 231

12 × 22/7 × r² = 231

r² = 231 × 7/22 × 1/12

r² = 6.125

∴ r = 2.47 cm

Now, on substituting value of 'r' in equation 3, we get,

(2(2.47) + 3h)/(2(2.47) + 3h - 12) = 2849/2618

(4.94 + 3h)/(4.94 + 3h - 12) = 2849/2618

2618(4.94 + 3h) = 2849(4.94 + 3h - 12)

12932.92 + 7854h = 14074 + 8547h - 34188

7181 = 693h

h = 7181/693

∴ h = 10.36 cm