Gel-filled cylindrical sensor housing

Core stretch

A watertight sensor housing has an outer cylindrical envelope of radius 4 cm and height 10 cm. A downward-facing hemispherical recess of radius 2.5 cm opens at its base and floods freely when immersed. Behind the thin curved recess wall, the gel chamber has radius 2.5 cm and height 10 cm, less the hemisphere. Ignore the thickness of the recess wall and top cover. Use \(\pi=3.142\) and give decimal answers to 2 d.p.

1. Find the exposed lower area, comprising the annulus and curved recess.[2]
2. The chamber is filled to 90% of usable capacity. Find the gel volume.[2]
3. The complete sensor has mass 400 g. Saline has density \(1.03\text{ g/cm}^3\). Its maximum upward buoyancy equals the weight of saline displaced at full immersion. Determine whether it floats, allowing for the flooded recess.[2]