(c) hr2 \(\big(\sqrt3-\frac{π}{2}\big)\)
The bases of the three cylinders when placed as given are as shown in the figure :
Let the radius of the base of each cylinder = r cm.
We are required to find the volume of air.
Space left between the cylinders = Area of shaded portion x height of cylinder
Now, area of shaded portion
= Area of ΔABC – Sum of areas of sectors of the three bases ΔABC, as can be seen is an equilateral triangle of side 2r.
∴ Area of Δ ABC = \(\frac{\sqrt3}{4}\times(2r)^2=\sqrt3r^2\)
Area of (sector AEF + sector BED + sector CFD) (∴ sector angles ∠A = ∠B = ∠C = 60º)
= 3 x \(\frac{\mathrm60^o}{\mathrm360^o}\timesπr^2=\frac{πr^2}{2}\)
∴ Required volume = \(\big(\sqrt3r^2-\frac{π}{2}r^2\big)h=\big(\sqrt3-\frac{π}{2}\big)r^2h.\)