Question

A test tube has lower part hemispherical and upper part cylindrical with same radius. If
5159/6 cm³ of water is added, the test tube will be just completely filled. But if 2002/3 cm³ of
water is added, 5 cm of height will remain empty. Find the radius and height of the cylindrical
part​

Answer 1

The radius and height of the cylindrical part are 3.5 cm and 20 cm respectively.

Step-by-step explanation:

Let the radius of the lower hemisphere is r cm and the height of the cylindrical part with radius r cm is h cm.

So, from the given conditions, we can write two different equations:

$\pi r^{2}h + \frac{2}{3}\pi r^{3} = \frac{5159}{6}$

⇒ $\pi r^{2}(h + \frac{2}{3}r) = \frac{5159}{6}$ .............. (1)

And, $\pi r^{2}(h - 5) + \frac{2}{3}\pi r^{3} = \frac{2002}{3}$

⇒  $\pi r^{2}(h - 5 + \frac{2}{3}r) = \frac{2002}{3}$ ............. (2)

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

$5\pi r^{2} = 192.5$

⇒ r = 3.5 cm.

Now, from equation (1) we get,

$\frac{22}{7}(3.5)^{2}h + \frac{2}{3} (\frac{22}{7})(3.5)^{3} = \frac{5159}{6}$

⇒ h = 20 cm

Therefore, the radius and height of the cylindrical part are 3.5 cm and 20 cm respectively. (Answer)