Competition problem — Geometry — Maths Olympiad Prep

Library / /26 of 94

Geometry Difficulty 4.5 AIME Prove it United States

Problem:

A balloon that blows up in the shape of a perfect cube is being blown up at a rate such that at time $t$ fortnights, it has surface area $6 t$ square furlongs. At how many cubic furlongs per fortnight is the air being pumped in when the surface area is $144$ square furlongs?

Solution

Solution:

The surface area at time $t$ is $6 t$ , so the volume is $t^{3/2}$ . Hence the air is being pumped in at a rate of $\frac{3}{2} \sqrt{t}$ . When the surface area is $144$ , $t=24$ , so the answer is $3 \sqrt{6}$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.