Question

A particle moves along the parabola y equals x squared in the first quadrant in such a way that its​ x-coordinate (measured in​ meters) increases at a steady 10 StartFraction m Over sec EndFraction . How fast is the angle of inclination theta of the line joining the particle to the origin changing when x equals 1 m question mark

Answer 1

The angle of inclination is changing at a rate of 2 radians per meter.

Step-by-step explanation:

To find the rate at which the angle of inclination is changing when x equals 1 m, we can use the concept of the tangent of the angle. The tangent of an angle is equal to the slope of the line that forms that angle.

Since the particle moves along the parabola y = x^2, the slope of the line joining the particle to the origin can be found by taking the derivative of y with respect to x.

Let's find the derivative of y = x^2:

Using the power rule, differentiate each term: dy/dx = 2x.

When x equals 1 m, the slope of the line is equal to 2(1) = 2.

Therefore, the angle of inclination theta is changing at a rate of 2 radians per meter.

Answer 2

(dθ/dt) = 1 rad / s = 57.3° /s

Step-by-step explanation:

- A particle moves along curve with function:

y = x^2

- The rate of change of x-coordinate is given by dx/dt = 10 m/s

Find:

How fast is the angle of inclination theta of the line joining the particle to the origin changing when x equals 1 m

Solution:

- The gradient of the line from origin to particle at position ( x , y ) is given by:

tan ( θ ) = y / x

Where, θ is the angle between x-axis and line from origin

x & y are coordinate of the point on given graph.

- To develop a rate of change expression we will derivate the above expression by time t:

d / dt (tan ( θ )) = d/dt (y / x )

(dθ/dt) / cos^2(θ) = (dy/dt) / (dx/dt)

(dθ/dt) = cos^2(θ) * (dy/dt) / (dx/dt)

- The rate of change of angle (dθ/dt) is given by above expression.

- We will apply the following chain rule to evaluate (dy/dt):

(dy/dt) = (dy/dx) * (dx/dt)

(dy/dt) = 2x * (10)

(dy/dt) = 20*x

@ x = 1, (dy/dt) = 20 m/s

@ x = 1, y = (1)^2 = 1

tan (θ) = 1

θ = 45°

- Now use the derived rate of change of angle expression we get:

(dθ/dt) = cos^2(45) * 20 / 10

(dθ/dt) = 0.5 * 20 / 10

(dθ/dt) = 1 rad / s = 57.3° /s