INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

June 23, 2020

Shortest Distance | PRMO II 2019 | Question 27

Try this beautiful problem from the Pre-RMO II, 2019, Question 27 based on Shortest Distance.

Shortest Distance - Pre-RMO II, Problem 27


A conical glass is in the form of a right circular cone. The slant height is 21 and the radius of the top rim of the glass is 14. An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If d the shortest distance it should crawl to reach the honey drop, what is the integer part of d?

Shortest Distance
  • is 107
  • is 36
  • is 840
  • cannot be determined from the given information

Key Concepts


Equation

Algebra

Integers

Check the Answer


Answer: is 36.

PRMO II, 2019, Question 27

Higher Algebra by Hall and Knight

Try with Hints


Rotate \(\Delta\)OAP by 120\(^\circ\) in anticlockwise then A will be at B, P will be at P'

Shortest Distance figure

or, \(\Delta\)OAP is congruent to \(\Delta\)OBP'

or, PB+PA=P'B+PB \(\geq\) P'P

Minimum PB+PA=P'P equality when P on the angle bisector of \(\angle\)AOB

or, P'P=2(21)sin60\(^\circ\)=21\(\sqrt{3}\)

[min(PB+PA)]=[21\(\sqrt{3}\)]=36 (Answer)

Subscribe to Cheenta at Youtube


Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
enter