Any naked eye would be able to tell that ~3.5 mm was way below the requisite 2 cm of oil.

In any case ... you know ... now that Miss Loi is back in Singapore, she can report that the massage was absolutely lifeless at the beginning of the extended period.

So annoyed was she that she was about to make a complaint when the phone rang again to inform the masseur something about some big-shot canceling his spa appointment ...

... and then all was well again

Suddenly she realized that despite the dimensions of the water and empty space having changed when the jar was flipped upright, their volumes remained the same.

At this point mathslover's approach of calculating and using the absolute volumes of the empty space & cone is safe and sound provided they can be calculated (which in this question they are).

Miss Loi shall provide an alternative approach just in case you meet a similar question that is evil enough not to provide any means of obtaining the absolute volumes (or areas for that matter):

When the jar is inverted,

⇒ V_container = 8 × V_water
⇒ V_{empty space} = V_container − V_water
= (8 × V_water) − (1 × V_water)
= 7 × V_water

When the jar is upright,

And now for Nona's moment of truth *teng teng teng teng*

∴ h = 0.3483 ≈ 0.348 cm (3 sig. fig.)

...

Gone were the slow and steady, rhythmic strokes that sent Miss Loi to a deep slumber in the past hour. In fact they now resembled the noob stuff one gets from a $16/40 min 'promotion' at People's Park Complex

For despite the blatantly low level of oil left in the jar, Nona, like a stubborn student who persistently hides from reality and is not being honest with him/her academic situation, refused to believe it to be <2 cm.

As if on cue, a breeze suddenly blew in from the sea, flipping the pages of Miss Loi's exam papers lying on a side table till they settled on the Mensuration and Similarity topics.

They revealed to Nona that the volume of a cone V_cone is often given in the formula sheet as

V_cone = --- (1)

and that, for some reason, cones in mensuration questions often involve playing around with the ratios of the height/volume/areas of similar figures.

1. And suddenly, through similar triangles, she was able to obtain the radius r_oil of the inverted cone of oil in Part (a)(i) as

   

   r_oil = = 2.5 cm

   Acknowledging that her math foundation is not strong, she carefully drew out the similar triangles (see diagram above) so as to avoid typical careless mistakes like the one below:

   

   The joy of solving her first ever mensuration problem was indescribable as she excitedly sub in r_oil = 2.5 cm and height h = 4 cm into the cone volume formula in (1) to obtain

   V_oil = cm³

   *Of course you may use the bigger similar triangles in mathslover's method but then you'll have to remember to divide the obtained diameter by two to get the final radius, and ensure you don't somehow use the wrong value in later parts which may result in kua kua kua careless mistake like the one committed by clarion above *kua kua kua*

2. The pages flipped again and revealed to Nona that the curved surface area of a cone is also given in the formula sheet as

   Curved surface area of cone = --- (2)

   The only thing that's missing here is the slant height l, and Nona applied with glee the Pythagoras Theorem she learnt in Sec One at her village school that allowed her to quickly obtain:

   

   l = √(4² + 2.5²) = 4.717 cm

   And sub in l = 4.717 cm into (2), she happily obtained the answer to Part (a)(ii):

   Surface area of jar in contact with oil
   = π × 2.5 cm × 4.717 cm
   = 37.047 cm²
   ≈ 37.0 cm² (3 sig. fig.)

...