How do you solve A subway track must pass through a cylindrical tunnel The tunnel is 6 m in diameter How wide should the track bed be so that the maximum height at the centre of the tracks is 4.5?

Answer 1

Draw a vertical diameter of a circle of 6m diameter. At a point 4.5 m from the top (T) draw a chord perpendicular to the diameter. Let the ends of the chord be A and B and the centre of the circle be O and the centre of the chord be C.

Then ∠AOB

=

2∠ATB

(the angle at the centre is twice the angle at the circumference).

If we just look at (say) the left hand side of this figure then ∠AOC

=

2∠ATC

Tan AOC =

AC/1.5 and Tan ATC =

AC/4.5.

Using the Identity, tan2A =

2tanA / (1 - tan2A)

Thus, AC/1.5 =

2AC/4.5(1 - [AC/4.5]2)

(1 - [AC/4.5]2) =

2/3

[AC/4.5]2) =

1/3

AC2 =

20.25/3 =

6.75

AC =

√6.75 =

2.60 (2dp)

Thus AB =

2AC =

5.20 m....which

is the width of the track bed.