A hemispherical bowl of internal radius 9 cm is full of liquid. It is to be filled into cylindrical bottles each of radius 1.5 cm and height 4 cm. How many bottles are needed?
54 bottles
Step 1: Write the formula for the volume of a hemisphere.
Volume of hemisphere = \( \dfrac{2}{3} \pi r^3 \)
Here, radius \( r = 9 \; \text{cm} \).
Step 2: Calculate the volume of the hemispherical bowl.
\( \dfrac{2}{3} \pi (9)^3 = \dfrac{2}{3} \pi (729) \)
\( = 486 \pi \; \text{cm}^3 \)
So, the hemispherical bowl can hold \( 486 \pi \; \text{cm}^3 \) of liquid.
Step 3: Write the formula for the volume of a cylinder.
Volume of cylinder = \( \pi r^2 h \)
Here, radius \( r = 1.5 \; \text{cm}, \; h = 4 \; \text{cm} \).
Step 4: Calculate the volume of one bottle (cylinder).
\( \pi (1.5)^2 (4) = \pi (2.25)(4) \)
\( = 9 \pi \; \text{cm}^3 \)
So, each bottle can hold \( 9 \pi \; \text{cm}^3 \) of liquid.
Step 5: Find the number of bottles required.
Total liquid in bowl ÷ liquid in one bottle
\( \dfrac{486 \pi}{9 \pi} = 54 \)
Final Answer: 54 bottles are needed.