A filling machine and an insufficient tail
(i) A certain machine A pumps liquid into an empty container of total volume . The first pumping action fills it with . Each subsequent action fills it with more than the previous action. Pumping continues until it is completely filled. Find the overflow volume at the final pumping action. marks
(ii) Suppose after A’s 50th action, a different machine B fills the remaining volume. B first fills , then each subsequent action fills the amount filled in the previous action.
(a) Find the total volume filled after B’s 10th action. marks
(b) Explain whether the container would be completely filled eventually. marks
[2013/CJC/Prelim/II/1]
Solution
Overflow at the final pump
Find the overflow when machine A first fills the container.
Hint
Use cumulative volume, not the amount in the final action.
Answer · (i)
overflow on action .
Sum the actions
Action adds cubic metres. Sum the actions to find the total volume.
Find the first filling action
On the source calculator, use Apps → PlySmlt2 → Polynomial Root Finder, choose order 2, and enter coefficients . The roots are about and ; reject the negative root because only positive integer action counts make physical sense.
Check the boundary and excess
Action 51 is still below capacity; action 52 exceeds it.
Ten actions by B
After 50 actions by A, find the total volume after ten actions by B.
Hint
Restart B’s index at 1 and add its sum to A 50.
Answer · (ii)(a)
.
Keep A’s existing liquid
A has already supplied 812.5 cubic metres.
Sum B separately
The new GP has first 5 and ratio 5/6; it does not continue A’s sequence.
Combine and round last
The source reports 838 cubic metres to 3 significant figures. Its displayed fraction is only a close rational approximation to B's addition; the exact sum is . Retain this exact formula until the final rounding.
Can B finish filling?
Explain whether B will ever finish filling the container.
Hint
Compare B’s full infinite sum with 37.5, not with 850.
Answer · (ii)(b)
Never: limiting total .
Calculate B’s ceiling
Every finite partial sum is below this limit because its terms are positive.
Compare with the remaining capacity
The shortage stays positive even at the limit.
Explore the idea
Separate cumulative volume from each pump action
Even infinitely many actions cannot add more than B’s convergent total.