Suppose that a camera is fixed at (0,0) in the coordinate plane (measured in feet). An actor starts at (10,10) and moves down (negative y-direction) at 1 foot per second, and moves right (positive x-direction) at 1 foot per second, so that his position at time t is (10+t,10−t).
Let θ be the angle between the positive x-direction and the line of sight from the camera to the actor as a function of t. Find the rate of change of θ as a function of time t.
(Type ∗ for multiplication; / for division; ∧ for exponentiation. The functions sqrt(x), ln(x), sin(x), etc. are known. Type e and pi for the mathematical constants e and π.)
dθ/dt = ?
please help me that I don't know how to start? Thank you in advance!
at time t,
tanθ = y/x = (10-t)/(10+t)
sec^2θ dθ/dt = -20/(10+t)^2
dθ/dt = -20/(10+t)^2 * 1/(1 + tan^2θ)
= -20/(10+t)^2 * 1/(1 + ((10-t)/(10+t))^2)
= -10/(t^2+100)
To find the rate of change of θ as a function of time t, we need to consider the relationship between the position of the actor and the line of sight from the camera.
The line of sight from the camera to the actor can be represented by a vector pointing from the camera (0, 0) to the actor's current position (10 + t, 10 - t). Let's denote this vector as r.
We can express this vector r as:
r = (10 + t, 10 - t)
The angle θ between the positive x-direction and the line of sight from the camera to the actor can be found using the dot product formula:
cos(θ) = (r · i) / ||r||
Here, r · i represents the dot product of the vector r and the unit vector i (representing the positive x-direction), and ||r|| represents the magnitude (or length) of the vector r.
First, let's find the dot product r · i:
r · i = (10 + t, 10 - t) · (1, 0) = (10 + t) * 1 + (10 - t) * 0 = 10 + t
Next, let's find the magnitude of the vector r:
||r|| = sqrt((10 + t)^2 + (10 - t)^2)
Now, we can substitute these values into the formula cos(θ) = (r · i) / ||r||:
cos(θ) = (10 + t) / sqrt((10 + t)^2 + (10 - t)^2)
To find the rate of change of θ with respect to time (dθ/dt), we need to differentiate this equation with respect to t. However, it's easier to work with sin(θ) rather than cos(θ). We can use the identity sin^2(θ) + cos^2(θ) = 1 to rewrite the equation:
sin^2(θ) = 1 - cos^2(θ)
=> sin^2(θ) = 1 - [(10 + t) / sqrt((10 + t)^2 + (10 - t)^2)]^2
Now, we can differentiate both sides with respect to t:
2 * sin(θ) * cos(θ) * dθ/dt = 2 * [(10 + t) / sqrt((10 + t)^2 + (10 - t)^2)] * [1 / sqrt((10 + t)^2 + (10 - t)^2)] * [(10 + t) - (10 + t)] * (1) + [(10 + t) / sqrt((10 + t)^2 + (10 - t)^2)]^2
Simplifying the equation:
2 * sin(θ) * cos(θ) * dθ/dt = [(10 + t) / sqrt((10 + t)^2 + (10 - t)^2)]^2
Finally, we can solve for dθ/dt by dividing both sides by 2 * sin(θ) * cos(θ):
dθ/dt = [(10 + t) / sqrt((10 + t)^2 + (10 - t)^2)]^2 / [2 * sin(θ) * cos(θ)]
To find the rate of change of θ as a function of time t, we need to calculate dθ/dt.
The line of sight from the camera to the actor is given by the vector (x, y) = (10+t, 10−t) - (0, 0) = (10+t,10−t).
The direction vector of this line of sight is defined as parallel to the line connecting the camera and the actor. We can normalize this vector to obtain the unit vector (u_x, u_y).
The dot product of this unit vector and the positive x-direction unit vector (1, 0) will give us the cosine of the angle θ between the positive x-axis and the line of sight.
Let's calculate the dot product:
(u_x, u_y) • (1, 0) = u_x • 1 + u_y • 0 = u_x.
Since both vectors are unit vectors, the magnitude of (u_x, u_y) will also be 1. Therefore, cos(θ) = u_x.
To find dθ/dt, we need to differentiate cos(θ) = u_x with respect to t.
Differentiating both sides, we have:
-d(sin(θ))/dt = du_x/dt.
To compute du_x/dt, we differentiate the unit vector (u_x, u_y) with respect to t.
u_x = (10+t) / √((10+t)² + (10-t)²) = (10+t) / √(200 + 20t).
Now, we differentiate u_x with respect to t:
du_x/dt = (1) / √(200 + 20t) - (10+t)(1/2)(200 + 20t)^(-3/2)(20).
This simplifies to:
du_x/dt = 1 / √(200 + 20t) - (10+t)(20) / (200 + 20t)^(3/2).
Now, substitute back into the equation -d(sin(θ))/dt = du_x/dt:
-d(sin(θ))/dt = 1 / √(200 + 20t) - (10+t)(20) / (200 + 20t)^(3/2).
Since d(sin(θ))/dt = cos(θ) • dθ/dt, we can rewrite the equation:
cos(θ) • dθ/dt = -1 / √(200 + 20t) + (10+t)(20) / (200 + 20t)^(3/2).
Now, we can solve for dθ/dt by dividing both sides by cos(θ):
dθ/dt = (-1 / √(200 + 20t) + (10+t)(20) / (200 + 20t)^(3/2)) / cos(θ).
Finally, substitute back in cos(θ) = u_x:
dθ/dt = (-1 / √(200 + 20t) + (10+t)(20) / (200 + 20t)^(3/2)) / (10+t) / √(200 + 20t).
Simplifying further:
dθ/dt = (-1 + (10+t)(20) / (200 + 20t)) / (10+t).
And that's the final rate of change of θ as a function of time t, dθ/dt.