SOLUTION: Two pulleys are connected by a belt. The radii of the pulleys asre 3 cm and 15 cm, and the distance between their centers is 24cm. Find the total length of belt needed to connect t

Question 452372: Two pulleys are connected by a belt. The radii of the pulleys asre 3 cm and 15 cm, and the distance between their centers is 24cm. Find the total length of belt needed to connect the pulleys.
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
Two pulleys are connected by a belt. The radii of the pulleys are 3 cm and 15 cm, and the distance between their centers is 24cm. Find the total length of belt needed to connect the pulleys.
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The key here is to note that the point at which the belt just leaves or enters the pulleys, is a point of tangency, that is, the line connecting these points on the two pulleys are at right angles to the radii of both pulleys. Call this line x for my following calculations. Draw a line starting from the center of the 3 cm pulley parallel to the tangent line until it intersects with the 15 cm radius of the larger pulley. You now have a right triangle with legs x and 12, with hypotenuse the given distance between the centers of the pulleys=24. You now can calculate x with the pythagorean theorem or a trig function.
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All you need to do next is calculate the rest of the belt that remains on the pulleys.
By using the x-12-24 right triangle, you can determine the central angles of the arcs of each pulley that the belts are on. Note for the small pulley the central angle is <180�, and >180� for the larger pulley. That part of the belt remaining on the pulleys can now be calculated by dividing the central angle by 360� times the circumference of the respective pulleys. Calculations follow:
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x^2=24^2-12^2=576-144=432
x=√432=20.78 cm
Working with 15 cm pulley:
cosA=12/24=1/2
A=60�
Central angle of arc=360-2*60=360-120=240�
Circumference=2π15=94.23
Part of belt on larger pulley=(240/360)*94.3=62.87 cm
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Working with 3 cm pulley:
B=90-A=30�
Central angle=360-2*90-2*30=120�
Circumference=2π3=18.85
Part of belt on smaller pulley=(120/360)*18.85=6.28 cm
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ans:
Total length of belt=2*20.78+62.87+6.28=110.71 cm
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