Footbridges Across a Circular Pond
A circular pond in a botanical garden has two straight wooden footbridges built as chords. The longer bridge AB is 48 m long and the shorter CD is 40 m long. They are parallel, lie on opposite sides of the centre, and the perpendicular distance between them is 22 m. The garden committee wants to add a third parallel bridge EF, 30 m long, somewhere between the two existing bridges. The structural engineer warns that any new bridge must have at least 6 m of clear horizontal separation from an existing parallel bridge to allow for abutments. A committee member argues that a 30 m bridge will easily fit because 30 m is less than both existing spans, and the 22 m gap looks wide. The gardener also needs to know the pond's radius to order a new floating boom that must reach from the centre to the edge.
1. Find the radius of the pond and the perpendicular distances of the two existing bridges from the centre.
Read the worked answer
Let the distances from the centre to AB and CD be $x$ and $y$, with $x+y=22$.
Half of AB is 24 m, so $r^2=x^2+24^2$. Half of CD is 20 m, so $r^2=y^2+20^2$.
Equating:
$$x^2+576=y^2+400 \Rightarrow y^2-x^2=176.$$
Factor: $(y-x)(y+x)=176$. Since $x+y=22$,
$$y-x=\frac{176}{22}=8.$$
Solving $x+y=22$ and $y-x=8$ gives $y=15$, $x=7$.
Then $r^2=7^2+24^2=49+576=625$, so $r=25$ m. The bridges are 7 m and 15 m from the centre.
2. Determine whether the proposed 30 m bridge can be placed between the two existing bridges while satisfying the 6 m separation rule.
Read the worked answer
For a chord of length 30 m, half the chord is 15 m. If its distance from the centre is $d$, then
$$d^2+15^2=25^2 \Rightarrow d^2=625-225=400,$$
so $d=20$ m.
The existing chords are at 7 m and 15 m on opposite sides of the centre. The region between them corresponds to distances from 7 m on one side to 15 m on the other; a chord at 20 m is not inside that 22 m gap. It lies 5 m beyond the 15 m chord.
On that side the separation is $20-15=5$ m, which is less than the required 6 m. On the opposite side the separation from the 7 m chord would be $20-7=13$ m, which is acceptable, but that is outside the gap. Therefore the bridge cannot be placed between the existing bridges.