A wheel with a diameter of 102cm has its centre 60 cm above the ground. It rotates once every 20 seconds. Point P will reach its maximum heights in 7 seconds.

Question

Determine the next time that point P will reach a height of 12 cm above the ground.

Answer 1

draw a diagram

the wheel has a diameter of 102 cm and its center is 60 cm above the ground which means that the wheel is 9 cm above the ground

it rotates every 20 seconds and the circumference is C=∏d, C=∏*102=320.44 cm

it takes 10 seconds to go from the bottom to the top and 10 seconds to go from the top to the bottom

the wheel travels 16.022 cm per second

in 7 seconds the wheel travels 112.154 cm

half the circumference is 160.22 cm

the wheel is already 9 cm above the ground

the point has to be at some place on the circle so that if a line is drawn straight down it will be 12 cm long

go up 3 cm from the bottom of the circle and draw a line across to the left until the line hits the circle

this is the point that you want that will be 12 cm above the ground

draw a radius to this point

you have 2 triangles

the first has a hypotenuse of 51, a leg of 48 (51-3) and the unknown line that you just drew

use Pythagorean Theorem to find that the line that you drew is 17.2336 cm long

the second triangle has legs of 3 and 17.2336 and an unknown hypotenuse-the length of the chord connecting the endpoints of the radii

use the Pythagorean Theorem to find the hypotenuse (chord length) to be 17.4928

now you have a third triangle which is isosceles when you connect the two endpoints of the two radii, the radius that you just drew and the vertical radius from the bottom of the circle to its center

draw the central angle bisector which is perpendicular to the base of the isosceles triangle

call the central angle x and half of the angle x/2

now you have a right triangle with hypotenuse 51 and one leg 17.4928/2=8.7464

sin(x/2)=8.7464/51

sin(x/2)=0.1715

use arcsin calculator to get x/2=9.875°

x=19.75°

(19.75/360)*320.44=17.579 cm

this is the rest of the arc length the point must travel after it goes from top to bottom to get to the point that is 12 cm above the ground

the point travels 10 seconds from top to bottom and then travels another amount of time

the point travels at 16.022 cm per second

17.579/16.022=1.097 seconds more the point must travel

total time the point travels is 10+1.097=11.097 seconds and it will be 12 cm above the ground