hey this is presh talwalkar
dear presh i am a high school student in
korea i greatly enjoy watching your
youtube channel
i'm writing to share with you a math
problem from the korean sat
called csat
college scholastic ability test
this problem is from the 1997 csat a
test that is known for having harder
questions than other years
i'm curious to know how you might solve
it thanks in advance
hyunchun
i try to pronounce every name correctly
but if i didn't get it correctly we can
say hj
this email and problem have been
slightly edited for presentation
purposes
so here's the problem
the diagram illustrates a right circular
cone-shaped mountain
if you build a shortest distance track
for a sightseeing train around the
mountain
in which the track starts at point a and
ends at point b
the track will first go uphill but then
it will go downhill
what is the length of the downhill track
wow
even this problem is pretty complicated
to understand
but lucky for you it's multiple choice
so there are four possible answers
200 divided by the square root of 19
300 divided by the square root of 30
300 divided by the square root of 91 and
400 divided by the square root of 91.
so i will say this problem was very
challenging for me
but
can you figure it out
give this problem a try and when you're
ready keep watching the video for a
solution
so this problem took me at least five
attempts before i solved it
there is no way i would have solved this
problem in a test
setting and here are some of the failed
attempts
i first thought the cone unwrapped into
a triangle
next i use coordinate geometry but i
failed to put the points a and b in the
correct positions
the third mistake is then i then made
many mistakes in trying to calculate the
equations of the lines
this led to many messy calculations
the fourth mistake is that i messed up
how to find the length of the downhill
track because i try to do it by
bisecting the line a b
this is not correct
i then tried to solve it by bisecting
the circular arc to find the downhill
track
so i was stuck and then i said
maybe i should try and find the answer
online but then suddenly it hit me
eventually i figured it out
i then also found an easier method to
solve the problem which i am presenting
to you
the coordinate method is linked in the
description
so why am i sharing this story with you
the lesson is that don't give up if you
can't solve a problem
and also don't peek at the video
solution too early
challenging problems often require
persistence but with practice and by
solving problems unmind your decisions
you will get better and you will become
great at math
so
how do we solve this problem
the key insight is that a right circular
cone unwraps into a circular arc
so here we have the cone-shaped mountain
and let's just imagine a general cone
if you cut open a right circular cone
it will unwrap into a circular arc
so here's a visualization of what i'm
talking about
so we have this cone
and it's a three-dimensional object
that makes the problem a little hard to
visualize
it'll be a lot easier if we can make it
two-dimensional
so let's imagine we're cutting this cone
vertically in the front
what will happen if we unwrap the cone
wow
so this cone turns into a perfect
circular arc
now why is it a circular arc
the reason is that each point on the
cone's base is the same distance of 60
from the cone's vertex
so here's the cone's base and at the top
is the cone's vertex
on the circular arc we have the cones
base at the bottom
and the cones vertex at the top
now each line
on this cones vertex to base this slant
height
exactly corresponds to a radius of this
circular arc
so that's why we end up with a circular
arc because every distance from the
vertex to the base is the same slant
height of 60.
and so this circular arc has a radius of
60.
so now how can we use this information
to continue to solve the problem
well the first step is we're going to
solve for the central angle of this
circular arc
the radius of the cone's base is 20.
that means its circumference is 2 pi r
which is 2 pi times 20
which is 40 pi
so that's the length of the circular arc
now we know the radius of the circular
arc is 60 this is capital r or the slant
height of the cone
now there's a formula
that relates the radius of a circular
arc its central angle to its length
so the length of the circular arc is r
times theta
so here we know the length is 40 pi
and r times theta is 60 theta
so we can solve that theta is equal to 2
pi over 3.
so that's the central angle
next we'll place points a and b on the
circular arc so what happens when we
unwrap this cone into a circular arc
well point a is along the base
so we know it'll have to be somewhere
along the base of this circular arc at
the very edge point b is 10 units from
the base and the track goes around the
cone
so point b is going to be on the
opposite side of where we place point a
so let's place point a on the right hand
side of the circular arc
we want to place point b on the other
side of the circular arc and 10 units
away from the base
so it'll be 10 units from the end of the
arc and it'll be 50 units from the
vertex because the radius is 60.
now what's the shortest path from a to b
well in this circular arc the answer is
simple
it's a straight line between points a
and b
so now
we're going to solve for the length of a
b
we draw a straight line between a and b
so how can we solve for this length
we can use a little bit of trigonometry
let's consider this triangle
we know two of its lengths and we know
the angle between those sides
so we can use the law of cosines to
figure out the length of the third side
c squared is equal to a squared plus b
squared minus two a b times cosine of
capital c the angle between those two
sides
so here we have the length of a b
squared is equal to 60 squared plus 50
squared minus 2 times 60 times 50 times
the cosine of the central angle 2 pi
over 3.
we can simplify this to get a b squared
is equal to 9 100 which means a b is
equal to 10 times the square root of 91.
so we're making good progress
but now
what is the downhill portion of the
track we know the total length of the
track but we're asked to figure out the
downhill portion
so the uphill part of a b
is increasing distance from the cone's
base
which is equivalently decreasing
distance to the cone's vertex
the downhill part of a b
is decreasing distance from the cone's
base which is equivalently increasing
distance to the cones vertex
so if we draw
subslant heights
which will be represented as radii
on this circular arc
the uphill portion
will be decreasing distance to the cones
vertex
and then the downhill part will be
increasing
furthermore the downhill part of a b
begins when the line is closest to the
cones vertex and that will be
perpendicular to a b
so this will be the point at which the
track goes from going uphill to going
downhill
now
we are already given that the track goes
from uphill to downhill and i use that
information to then say we'll look for
the perpendicular and this has to be the
point
but to truly prove this problem for
extra credit you can prove that the
track goes uphill and then downhill and
i'm sure some of you who think all these
problems are easy will be able to figure
this out you know it won't be very
difficult for you but it's actually a
little bit of a complicated thing to
show
anyway this is a test question we're
going to use this information that was
given to us and then continue to solve
the problem
so now how do we figure out the downhill
length
we again
use some geometry
let's label the downhill portion x
the uphill portion will be
10 square root of 91 minus x
and let's label
the distance from the vertex to the line
a b as h
by the pythagorean theorem in one
triangle the uphill portion
we have the quantity 10 times the square
root of 91 minus x squared plus h
squared is equal to 60 squared and in
the downhill triangle we have x squared
plus h squared is equal to 50 squared
now there's a neat trick we can subtract
the second equation from the first and
we'll cancel out the h squares and we'll
also cancel out the x squares
so this will simplify to be
9 100 minus 2
times 10 times the square root of 91
times x is equal to 60 squared minus 50
squared
we can simplify this equation and then
solve to get x is equal to 400 divided
by the square root of 91.
and that's it that's our answer so the
correct answer is 4
400 divided by the square root of 91.
so this is an incredible test question
because there are so many concepts that
you need to know in order to solve this
correctly
but it was very satisfying for me to
learn all these things about a cone how
to figure out the downhill portion the
uphill portion and i hope you also got
some of that discovery while solving
this problem
now did you figure it out and what
method did you use
thanks for watching this video please
subscribe to my channel i make videos on
math you can catch me on my blog mind
your decisions
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if you have a math problem or math topic
you can email me presh
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decisions or at presh talwalkar