We Consider a Triangle ABC with Fixed Base Length b = BC and the Area of the Triangle: Mathematics Homework, NUS, Singapore

Optimization Problem

1. We consider a triangle ABC with fixed base length b = BC and the area of the triangle. Prove that among all such triangles ABC, the one with the shortest perimeter length is the isosceles triangle with AB = AC.

2. During the wildfire, a huge amount of ash fell on the swimming pool of Mrs. M. Since the circulation pump continuously operates, the water has become murky with ash. Her swimming pool contains 108,000 liters of water, and its purifying system filters 10 liters of water per second with a powerful pump. The filter removes 100% of ash as water goes through the purifying system. The purified water is pumped back to the swimming pool and immediately mixed with the rest of the water.

After operating the purifying system for 3 consecutive hours, how much ash still remain in the water, compared to the initial amount? Give your answer in terms of percentage with 5 decimal places, such as 31:41592%.

Hint: This percentage is a universal constant, if the capacity of the pool c, the speed of the pump s, and the duration of operation to satisfy the relation t = c=s, same as in the above problem.