Marbles of diameter 1.4 cm are dropped into a cylindrical beaker of diameter 7 cm containing some water. Find the number of marbles so that the water level rises by 5.6 cm.
150 marbles
Step 1: Volume of water displaced (rise in cylinder).
The water level rises by 5.6 cm in the cylindrical beaker of diameter 7 cm.
Radius of beaker = \( \dfrac{7}{2} = 3.5 \) cm.
Volume of water displaced = volume of cylinder part filled = \( \pi r^2 h \).
\( V_{cylinder} = \pi (3.5)^2 (5.6) \).
\( V_{cylinder} = \pi \times 12.25 \times 5.6 = 68.6\pi \; cm^3 \).
Step 2: Volume of one marble.
Diameter of marble = 1.4 cm, so radius = \( 0.7 \) cm.
Volume of a sphere = \( \dfrac{4}{3}\pi r^3 \).
\( V_{marble} = \dfrac{4}{3}\pi (0.7)^3 \).
\( V_{marble} = \dfrac{4}{3}\pi (0.343) = 0.457\pi \; cm^3 \).
(In fraction form, \( V_{marble} = \dfrac{1372}{3000}\pi \; cm^3 \)).
Step 3: Number of marbles needed.
Total displaced volume must equal total volume of marbles.
So, Number of marbles = \( \dfrac{V_{cylinder}}{V_{marble}} \).
\( N = \dfrac{68.6\pi}{0.457\pi} \).
\( N = 150 \).
Final Answer: 150 marbles are required.