Two towns, A and B are separated by a straight highway of 60km. Town A has an abundance water supply and is able to share it to local communities via a pipeline. Town A is due west of town B and the pipeline leaves town A at N60degreesE.
Town B would like to get water from this pipeline, but has to build a 40km pipeline of its own due to contactual arrangements.
Will town B be able to connect to town A's pipeline? If so, at what distance(s) along town A's pipeline can they connect?
Town B would like to get water from this pipeline, but has to build a 40km pipeline of its own due to contactual arrangements.
Will town B be able to connect to town A's pipeline? If so, at what distance(s) along town A's pipeline can they connect?
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Is it OK to use Trigonometry? I'd have to think rather hard to work out a non-trig method.
Draw the line BC perpendicular to the pipeline from A, meeting the pipeline at C.
i.e. angle BCA is a rightangle.
Angle CAB = 30°
(I hope that's obvious: AC makes 60° with the line due north, so CAB = 90 - 60)
In the rightangled triangle CAB,
sin 30° = BC/AB
0.5 = BC/60
BC = 30.
So yes, a 40km pipeline from B will connect to the pipeline from A.
Call the point at which it connects P.
Then one method is to work out the size of angle PBC from the rightangled triangle PBC with one side 30 and hypotenuse 40.
Subtract angle PBC from 60° to find the size of angle ABP,
then apply the sine rule to triangle ABP to find the length of AP.
However I'm going to let AP = x and apply the cosine rule to triangle ABP. This will lead to a quadratic equation in x. It will have two solutions, because there are two possible positions for P: either between A and C as I've assumed in the above method outline, or on the far side of C.
Angle A = 30°
hence
cos 30° = (x² + 60² - 40²)/(2*60x)
x² - (120 cos30°)x + 2000 = 0
x² - (60√3)x + 2000 = 0
x = (60√3 ±√(10800 - 8000)) / 2
. = 30√3 ±√(700)
. = 25.5 or 78.4 correct to one decimal place.
Hence they can connect to A's pipeline 25.5 km along the line or 78.4 km along the line.
Draw the line BC perpendicular to the pipeline from A, meeting the pipeline at C.
i.e. angle BCA is a rightangle.
Angle CAB = 30°
(I hope that's obvious: AC makes 60° with the line due north, so CAB = 90 - 60)
In the rightangled triangle CAB,
sin 30° = BC/AB
0.5 = BC/60
BC = 30.
So yes, a 40km pipeline from B will connect to the pipeline from A.
Call the point at which it connects P.
Then one method is to work out the size of angle PBC from the rightangled triangle PBC with one side 30 and hypotenuse 40.
Subtract angle PBC from 60° to find the size of angle ABP,
then apply the sine rule to triangle ABP to find the length of AP.
However I'm going to let AP = x and apply the cosine rule to triangle ABP. This will lead to a quadratic equation in x. It will have two solutions, because there are two possible positions for P: either between A and C as I've assumed in the above method outline, or on the far side of C.
Angle A = 30°
hence
cos 30° = (x² + 60² - 40²)/(2*60x)
x² - (120 cos30°)x + 2000 = 0
x² - (60√3)x + 2000 = 0
x = (60√3 ±√(10800 - 8000)) / 2
. = 30√3 ±√(700)
. = 25.5 or 78.4 correct to one decimal place.
Hence they can connect to A's pipeline 25.5 km along the line or 78.4 km along the line.