Question

You are given a triangle of constant size (not changing over time), where one side is currently 20 cm long and increases at a rate of 3 cm per second, and a second side is currently 30 cm long and decreases at a rate of 2 cm per second. The angle α between the two sides is currently π/6. At what rate does α change?

Answer 1

The rate at which α changes can be determined using trigonometry. Using the given rates of change of the sides of the triangle, we can calculate the rate of change of α. The rate at which α changes is 10.1 cm/s.

Step-by-step explanation:

The rate at which α changes can be determined using trigonometry. We know that the tangent of α is equal to the length of the side opposite α divided by the length of the adjacent side. So we can take the derivative of the tangent function with respect to time to find the rate of change of α.

d(tan α)/dt = (d(20 cm)/dt)/(30 cm) - (20 cm)/(d(30 cm)/dt)

Using the given rates, we can substitute in the values and solve for d(tan α)/dt.

d(tan α)/dt = (3 cm/s)/(30 cm) - (20 cm)/(-2 cm/s) = 0.1 - (-10) = 10.1 cm/s