A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter; the diameter of the spherical part is 8.5 cm. By measuring the amount of water it holds, a child finds its volume to be 345 cm³. Check whether she is correct, taking the above as the inside measurements, and π = 3.14.

Given,

Diameter of sphere (D) = 8.5 cm

Radius of sphere (R) = $\dfrac{D}{2} = \dfrac{8.5}{2}$ = 4.25 cm.

Height of cylindrical neck (h) = 8 cm

Diameter of cylindrical neck (d) = 2 cm

Radius of cylindrical neck (r) = $\dfrac{d}{2} = \dfrac{2}{2}$ = 1 cm.

Volume of glass vessel = Volume of sphere + Volume of cylinder

$= \dfrac{4}{3}πR^3 + πr^2h \\[1em] = π\Big(\dfrac{4}{3}R^3 + r^2h\Big) \\[1em] = 3.14 \times \Big(\dfrac{4}{3} \times (4.25)^3 + 1^2 \times 8\Big) \\[1em] = 3.14 \times \Big(\dfrac{4}{3} \times 76.77 + 1 \times 8\Big) \\[1em] = 3.14 \times \Big(\dfrac{307.0625}{3} + 8\Big) \\[1em] = 3.14 \times (102.35 + 8) \\[1em] = 3.14 \times 110.35 \\[1em] = 346.51 \text{ cm}^3.$

∴ Volume of water in vessel = 346.51 cm^3.

Hence, volume of water in vessel is not equal to 345 cm³.