Question 5.6.9: Graphing calculator required A 20-inch belt connects a pulle...
Graphing calculator required
A 20-inch belt connects a pulley with a very small shaft on a motor, as shown in Fig. 5.91. The distance between the shaft and the large pulley is 4 inches. Assuming the shaft is a single point, find the radius of the large pulley to the nearest tenth of an inch.
Let x be the radius of the circle and w be the distance from the shaft to point A, as shown in the figure. Since the radius is perpendicular to the belt at A, we have \theta=\cos ^{-1}\left(\frac{x}{x+4}\right). Assuming θ is in radians, the length of the arc of the circle intercepted by θ is x \cdot \cos ^{-1}\left(\frac{x}{x+4}\right). Since the circumference of the circle is 2πx, the length of the long arc from A to B is
2 \pi x-2 x \cdot \cos ^{-1}\left(\frac{x}{x+4}\right)By the Pythagorean theorem, x^{2}+w^{2}=(x+4)^{2} or w=\sqrt{8 x+16}. Since the total length of the belt is 20 inches, we have
2 \pi x-2 x \cdot \cos ^{-1}\left(\frac{x}{x+4}\right)+2 \sqrt{8 x+16}=20One way to solve this equation with a graphing calculator is to graph
y=2 \pi x-2 x \cdot \cos ^{-1}\left(\frac{x}{x+4}\right)+2 \sqrt{8 x+16}-20and find the x-intercept, as shown in Fig. 5.92. The radius of the large pulley is approximately 2.2 inches.
