Anonymous
Anonymous asked in Science & MathematicsMathematics · 4 weeks ago

# A belt is stretched tightly over two wheels having a radii of 1 and 5 inches, respectively. If the center of the wheels...?

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are 8 inches apart, what is the total length of the straight sections of the belt?

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Look at the very helpful second diagram in the link given by RealPro

but replace 5cm, 3cm 14 cm by 5 inches, 1 inch, 8 inches

One straight section is given by L =√[8^2 – (5 – 1)^2] = 2√(7) inches

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• Refer to the 'rough' sketch below.

The belt touches the circumference of each wheel at points A and B. The lengths make up a trapezium. The parallel sides are the radii of 1 inch and 5 inches. The distance between them is AB and the slant side has length 8 inches.

From this, a right triangle is evident

so, using Pythagoras' theorem we have:

8² = (AB)² + (5 - 1)²

so, 64 = (AB)² + 16

=> (AB)² = 48

Hence, AB = √48 => 6.93 inches

so, straight sections total 2 x 6.93 = 13.9 inches

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r1 = 5 in

r2 = 1 in

d = 8 in

2sqrt( d^2 - (r1-r2)^2 ) + 2pi r1 + 2(r2-r1)arccos( (r1-r2)/d ) = 36.89 in

Output of arccos needs to be in radians.

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