A particle is moving along the graph of y=x^1/3.
Suppose x is increasing at the rate of 3 cm/s.
At what rate is the angle of inclination, theta, changing when x=8?
[Hint: when x=8, theta approx. 0.24 rad]Well, the derivative of a constant (0.24) is 0 so that wouldn't do a whole lot of good would it?NIZBIT said:
It's a lot more helpful to find the derivative of a function rather than a number! Draw a picture, showing the "angle of inclination" and write theta as a function of x. Then find the derivative of that function with respect to x. (Do you remember that the derivative of a function is the tangent of the angle the tangent line to the graph makes with the x-axis?) A particle moving along y=x^1/3 means that the position of the particle is determined by the function y=x^1/3, where x is the independent variable and y is the dependent variable.
The equation x=8 means that the particle is moving along the line x=8, which is a vertical line passing through the point (8,0). This indicates that the particle is constrained to move only along this line.
This means that the angle of rotation for the particle is approximately 0.24 radians, where 1 radian is equivalent to about 57.3 degrees. This angle is used to calculate the rate of change of the particle's position.
The rate of change of the particle's position can be calculated using the formula dy/dx = (dy/dt) / (dx/dt), where dy/dt and dx/dt are the rates of change of y and x respectively with respect to time.
Since the particle is moving along the line x=8, the rate of change of x (dx/dt) is constant. However, the rate of change of y (dy/dt) will vary depending on the angle of rotation (theta) and can be calculated using trigonometric functions such as sine and cosine.