Question

A physics experiment is set up using two pulleys, a string, and a weight as shown below. The larger pulley has a radius of 20 centimeters, and the smaller pulley has a radius of 5 centimeters. The distance between the centers of the pulleys is 51 centimeters. The string is pulled tightly across both pulleys so that AB is a common tangent of the pulleys. Find the length of string from A to B to the nearest tenth of a centimeter.

Answer 1

Where is the diagram?

Answer 2

We can use the relationship between the circumference of a circle and its radius to find the length of string from A to B.

Let C1 be the circumference of the larger pulley with radius 20 centimeters, and C2 be the circumference of the smaller pulley with radius 5 centimeters. We can use the formula C = 2πr to calculate these circumferences:

C1 = 2π(20) = 40π
C2 = 2π(5) = 10π

Next, we can use the Pythagorean theorem to find the length of the string from A to B. Let x be the length of the string from A to B. Then, we have:

x^2 = (51)^2 + (C1 - C2)^2
x^2 = 2601 + (40π - 10π)^2
x^2 = 2601 + (30π)^2
x ≈ 97.3

Rounded to the nearest tenth of a centimeter, the length of the string from A to B is approximately 97.3 centimeters.

Answer 3

\[x=\sqrt{51^2-15^2}=\sqrt{2601-225}=\sqrt{2376}\approx48.74 cm\]