Step-by-step explanation:
The distance between the two people is changing because one is walking away horizontally while the other is moving vertically. To find the rate at which the distance between them changes, we can use the Pythagorean theorem:
\[ \text{Distance}^2 = \text{Horizontal distance}^2 + \text{Vertical distance}^2 \]
Let's denote the horizontal distance by \( x \) (the person walking away) and the vertical distance by \( y \) (the person going up in the elevator).
So, after 10 seconds:
Horizontal distance, \( x = \text{speed} \times \text{time} = 1.2 \text{ feet/second} \times 10 \text{ seconds} = 12 \text{ feet} \)
Vertical distance, \( y = \text{speed} \times \text{time} = 7 \text{ feet/second} \times 10 \text{ seconds} = 70 \text{ feet} \)
Now, applying the Pythagorean theorem:
\[ \text{Distance}^2 = x^2 + y^2 = 12^2 + 70^2 \]
\[ \text{Distance} = \sqrt{12^2 + 70^2} \approx 70.8 \text{ feet} \]
Differentiating the equation \( \text{Distance}^2 = x^2 + y^2 \) implicitly with respect to time gives:
\[ 2 \times \text{Distance} \times \frac{d(\text{Distance})}{dt} = 2x \times \frac{dx}{dt} + 2y \times \frac{dy}{dt} \]
Given that \( \frac{dx}{dt} = 1.2 \text{ feet/second} \) and \( \frac{dy}{dt} = 7 \text{ feet/second} \), and solving for \( \frac{d(\text{Distance})}{dt} \):
\[ 2 \times 70.8 \times \frac{d(\text{Distance})}{dt} = 2 \times 12 \times 1.2 + 2 \times 70 \times 7 \]
\[ \frac{d(\text{Distance})}{dt} = \frac{24 \times 1.2 + 140 \times 7}{70.8} \approx \frac{168.8}{70.8} \approx 2.38 \text{ feet/second} \]
Therefore, the rate at which the distance between the two people is changing after 10 seconds is approximately \(2.38\) feet per second.
Answer: To find the rate at which the distance between the two people is changing, we can use the concept of relative motion.
Let's consider the person walking away from the elevator as Person A and the person going up the elevator as Person B.
The rate at which the distance between the two people is changing can be determined by finding the derivative of the distance function with respect to time.
Let's assume that the initial distance between Person A and Person B is D feet.
After 10 seconds, Person A would have walked a distance of 1.2 feet/second * 10 seconds = 12 feet.
Person B, moving up the elevator at a rate of 7 feet per second, would have traveled a distance of 7 feet/second * 10 seconds = 70 feet.
Therefore, after 10 seconds, the distance between Person A and Person B would be D + 12 feet + 70 feet = (D + 82) feet.
Taking the derivative of this distance function with respect to time, we get:
d(Distance)/dt = d(D + 82)/dt = 0 + 0 = 0
This means that the rate at which the distance between the two people is changing after 10 seconds is 0 feet per second.
Therefore, the distance between the two people is not changing after 10 seconds.
