Question

Please give answers of both with full explanation
Ch - Surface Areas And Volumes
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Answer 1

Question 1:

A solid right circular cylinder is made with clay whose diameter is 4 cm and height is 48 cm. Reema wants to make a certain number of spheres of diameter 4 cm. How many solid spheres can be made out of it?

Solution:

The main concepts to be used in this question are the volume formulae of cylinder and sphere. At first, we need to find the volume of the complete cylinder and then the volume of one sphere. Dividing these quantities, we will get the number of spheres.

⇒ Diameter of the clay cylinder = 4 cm

⇒ Radius = 2 cm

⇒ Height of the clay cylinder = 48 cm

Volume of a cylinder = πr²h

Substituting the values we know into the equation:

$\sf{=\pi\times2\times2\times\times48}$

$\sf{=192\pi}$

Please note that I have not expanded the value of π because we can cancel the values off later on.

⇒ Diameter of one sphere = 4 cm

⇒ Radius of one sphere = 2 cm

Volume of a cone = 4/3πr³

Substituting the values we know into the equation:

$\sf{=\dfrac{4}{3}\pi\:2\times2\times2}$

$\sf{=\dfrac{32\pi}{3}}$

Please note that I have not expanded the value of π because we can cancel the values off later on.

Now, the number of spheres that can be made:

$\sf{=\dfrac{Volume\:of\:cylinder}{Volume\:of\:one\:sphere}}$

$\sf{\dfrac{192\pi}{\cfrac{32 \pi}{3}}$

$\sf{=192\pi\times\dfrac{3}{32\pi}}$

$\sf{=6\times3}$

$\sf{=18}$

∴ The number of spheres that can be made are 18.

Question 2:

Water flows through a circular pipe whose internal diameter is 2 cm at the rate of 6 m/s into a cylindrical tank whose base radius is 60 cm. Find the rise in level of water in 30 minutes.

Solution:

The main concept used in this question is the volume of cylinder.

⇒ Inner diameter of pipe = 2 cm

⇒ Inner radius of pipe = 1 cm which can be written as 1/100 m.

As the water flows through the pipe at the rate of 6 m/s,

⇒ Height of the pipe = 6 m

Therefore, the volume of water that flows through the pipe in 1 second is:

= Volume of cylinder

= πr²h

$\sf{=\dfrac{22}{7}\times\dfrac{1}{100}\times\dfrac{1}{100}\times6\times30\times60}$

$\sf{=\dfrac{22}{7}\times\dfrac{1}{10000}\times6\times30\times60}$

This is exactly equal to the volume of water that rises in the cylindrical tank after 30 minutes.

Let the height of the tank be h.

⇒ Base radius of the tank = 60 cm = 60/100 m

Volume of cylinder = πr²h

From the given statement:

$\sf{=\dfrac{22}{7}\times\dfrac{60}{100}\times\dfrac{60}{100}\times\:h=\dfrac{22}{7}\times\dfrac{1}{10000}\times6\times30\times60}$

$\sf{=\dfrac{60\times60}{10000}\:h=\dfrac{6\times30\times60}{10000}}$

$\sf{h=\dfrac{3\times36}{36}}$

$\sf{h=3\:m}$

Hence the height to which the water rises is 3 m.

Answer 2

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