What scratch?
The diagram below shows a simplified side view of a garden roller. The radius of the cylinder is 30 cm. The length of the handle is 130 cm. The roller is leaning against a wall with the handle at an angle of 25 degrees to the vertical the ground is horizontal.
A gardener takes the roller on to a lawn. Just as he begins to roll he notices a small scratch at the highest point of the cylinder.
(iv) Show that, when the cylinder has been pushed a distance x cm on the lawn, the height of the scratch above the ground is:
30{1+cos(6x/pi)}cm
Anyone know how to get there?
If the roller has been pushed x cm along the ground, it has rotated by the x/(circumference) of a revolution: that is, by x/30 radians or 360x/(2.pi.30) degrees. So the height of the scratch above the level of the centre is now 30cos(60x/pi): draw a picture showing the roller rotated by 60x/pi. The centre is itself 30 cm above the ground.