Question

A cylindrical vessel of height $ 24\;cm $ and diameter $ 40\;cm $ is full of water. Find the exact number of small cylindrical bottles, each of height $ 10\;cm $ and diameter $ 8\;cm $ , which can be filled with this water.

Answer 1

Hint: Here we will find the volume of the bigger cylindrical vessel and the volume of the small cylindrical bottles, and then take the ratio of both the volumes to know the number of bottles filled. Use formula $ V = \pi {r^2}h $ to find the volume of the cylindrical vessels.

Complete step-by-step answer:
Measures of the bigger cylindrical vessel are given –
Height, $ h = 24\;cm $
Diameter, $ d = 40\;cm $
Convert the given diameter into radius. Radius is half of the diameter.
Place the values from the given data-
 $
  r = \dfrac{d}{2} \\
   \Rightarrow r = \dfrac{{40}}{2} \\
   \Rightarrow r = 20\;cm \;
  $
Now, the volume of the cylindrical vessel is
 $ V = \pi {r^2}h $
Place the values –
 $ V = \pi \times 20 \times 20 \times 24\;c{m^3} $ ...... (i)
Measures of the bigger cylindrical vessel are given –
Height, $ h = 10\;cm $
Diameter, $ d = 8\;cm $
Convert the given diameter into radius. Radius is half of the diameter.
Place the values from the given data-
 $
  r = \dfrac{d}{2} \\
   \Rightarrow r = \dfrac{8}{2} \\
   \Rightarrow r = 4cm \;
  $
Similarly the volume of the small cylindrical bottles is $ V = \pi {r^2}h $
Place the values in the above equations –
 $ V = \pi \times 4 \times 4 \times 10\;c{m^3} $ ...... (ii)
Now, the count of number of small cylindrical bottles filled $ = \dfrac{{Volume\,{\text{of bigger vessel}}}}{{Volume\;{\text{of small bottles}}}} $
Place the values in the above equations from (i) and (ii)
Number of bottles, $ N = \dfrac{{\pi \times 20 \times 20 \times 24}}{{\pi \times 4 \times 4 \times 10}} $
Common Multiples from the numerator and the denominator cancel each other, so remove from the numerator and the denominator.
 $ N = \dfrac{{20 \times 20 \times 24}}{{4 \times 4 \times 10}} $
Simplify the above equation –
 $ \Rightarrow N = 60 $
Hence, exactly $ 60 $ bottles can be filled from the given cylindrical vessel.
So, the correct answer is “60”.

Note: Always check the given data and convert the data as per requirement. As such we here converted diameter into radius. Also, remember when you are taking ratios the units of both the volumes should be in the same format cm or meter. Check the given units twice.