Two poles are 24 metres and 30 metres high and 20 metres apart. A slack wire joins the tops of the poles and is 32 metres long. A cable car is moving along the wire at 5 metres per second away from the shorter pole. When the car is 12 metres horizontally from the shorter pole, it is 15 metres high and the length of the wire to the shorter pole is 15 metres. At this instant, find: a) how fast the car is moving horizontally. b) how fast the car is moving vertically.

Question

Two  poles  are  24  metres  and  30  metres  high  and  20  metres  apart.    A  slack wire  joins  the  tops  of  the  poles  and  is  32  metres  long.    A  cable  car  is  moving along  the  wire  at  5  metres  per  second  away  from  the  shorter  pole.    When  the car  is  12  metres  horizontally  from  the  shorter  pole,  it  is  15  metres  high  and  the length  of  the  wire  to  the  shorter  pole  is  15  metres.    At  this  instant,  find: a)    how  fast  the  car is  moving  horizontally. b)    how fast  the  car  is  moving  vertically.

Steve · Answer

No ideas? OK. I'll get you started. As usual, with a diagram.

Let's set some labels:

S = top of short pole
T = top of tall pole
C = cable car
P = point on ground directly below the car
Draw a horizontal line from C which intersects the short and tall poles at Q and R, respectively.
B = bottom of short pole
D = bottom of tall pole.

x = BP = QC (hor. distance of car from short pole)
y = QS (distance of car below short pole)
RT = y+6 (distance of car below tall pole)
u = CS (length of cable from short pole)
CT = 32-u (cable length to tall pole)

That means we have at the moment in question,
x=12
y=9
z=15
u=15
v=17

Wow - notice how all the distance come out to be integers? Well crafted problem!

We want to find dx/dt and dy/dt.

x^2+y^2 = u^2
(20-x)^2 + (y+6)^2 = (32-u)^2

See what you can do with that.