F. (12) (1 punto posible)
You are at a point x=x(t) along a horizontal line, representing the ground. You are flying a kite which maintains a constant height of 40 meters. Assume also that the kite string is a straight line. The kite is above the point k=k(t) on the ground, which is currently 30 meters behind you.
You are running to the right at 1m/sec; in other words, x′(t)=1 m/s. Also, the kite string is lengthening at 1 m/sec. What is the horizontal velocity of the kite k′(t) at this moment?
(Type ∗ for multiplication, / for division, and ∧ for exponentiation; or enter as decimal to 2 decimal
To find the horizontal velocity of the kite, we need to differentiate the position function of the kite with respect to time.
We know that the position of the kite along the ground, k(t), is currently 30 meters behind you. Since you are running to the right at a rate of 1 m/s, your position function x(t) can be written as x(t) = t.
Given that the height of the kite is constant at 40 meters, the position function of the kite can be written as k(t) = x(t) - 30 = t - 30.
Now, let's differentiate k(t) with respect to time to find the horizontal velocity of the kite.
k'(t) = d/dt(t - 30)
= 1
Therefore, the horizontal velocity of the kite, k'(t), is 1 m/s.
To find the horizontal velocity of the kite, k'(t), we can start by understanding the relationship between the variables given.
Given:
- The kite is currently 30 meters behind you (k = k(t) = -30)
- You are running to the right at 1 m/s (x'(t) = 1)
- The kite string is lengthening at 1 m/s
Since the kite is maintained at a constant height of 40 meters, we can use similar triangles to relate the vertical position of the kite (k) to the horizontal position (x).
Let's set up a ratio using the similar triangles formed by the kite and the ground:
k / x = 40 / h
Where h is the height above the ground that the kite is at. Since the height is constant at 40 meters, we can substitute h = 40 into the equation:
k / x = 40 / 40
k / x = 1
Now, let's take the derivative of both sides with respect to time (t) to find the relationship between the rates of change:
k' / x' = 0
k' = 0
Since k' = 0, it means that the horizontal velocity of the kite, k'(t), is zero at this moment.
Therefore, the horizontal velocity of the kite is 0 m/s.