Question

A kite at a constant 25 meters above the ground begins to move horizontally further away from the person flying the kite. If l represents the length of the kite string and θ represents the angle between the string and the height of the kite, which of the following relates the rate at which the string lengthens with the rate at which the angle increases, with respect to time?

1) The rate at which the string lengthens is directly proportional to the rate at which the angle increases
2) The rate at which the string lengthens is inversely proportional to the rate at which the angle increases
3) The rate at which the string lengthens is constant and independent of the rate at which the angle increases
4) The rate at which the string lengthens is zero

Answer 1

The rate at which the string lengthens is directly proportional to the rate at which the angle increases.

Step-by-step explanation:

The rate at which the string lengthens and the rate at which the angle increases are related in this scenario. Since the kite is flying at a constant height above the ground, the increase in angle corresponds to the increase in horizontal distance between the person flying the kite and the kite itself. As the kite moves further away, the string lengthens and the angle increases. Therefore, the rate at which the string lengthens is directly proportional to the rate at which the angle increases.

This means that as the angle increases at a faster rate, the string lengthens at a faster rate as well. Similarly, if the angle increases at a slower rate, the string lengthens at a slower rate too.

Answer 2

The rate at which the string lengthens is directly proportional to the rate at which the angle increases.

Step-by-step explanation:

In this scenario, the rate at which the string lengthens (dl/dt) and the rate at which the angle increases (dθ/dt) are related. Since the kite is moving horizontally further away from the person flying it, the length of the string (l) increases. This means that dl/dt is positive. Additionally, as the kite moves further away, the angle (θ) between the string and the height of the kite also increases. This means that dθ/dt is also positive.

Based on the given information, we can conclude that the rate at which the string lengthens is directly proportional to the rate at which the angle increases. Therefore, the correct option is 1) The rate at which the string lengthens is directly proportional to the rate at which the angle increases.