Answer:
2.3
Step-by-step explanation:
![-5.3\:\text{ft/s}](https://tex.z-dn.net/?f=-5.3%5C%3A%5Ctext%7Bft%2Fs%7D)
Step-by-step explanation:
We start by applying the Pythagorean theorem to the ladder, with its length L as the hypotenuse:
![L^2 = 100\:\text{ft}^2 = x^2 + y^2](https://tex.z-dn.net/?f=L%5E2%20%3D%20100%5C%3A%5Ctext%7Bft%7D%5E2%20%3D%20x%5E2%20%2B%20y%5E2)
where x is the vertical distance from the top of the ladder to the ground and y is the horizontal distance from the bottom of the ladder to the wall. Taking the derivative of the above expression with respect to time, we get
![0 = 2x\dfrac{dx}{dt} + 2y\dfrac{dy}{dt}](https://tex.z-dn.net/?f=0%20%3D%202x%5Cdfrac%7Bdx%7D%7Bdt%7D%20%2B%202y%5Cdfrac%7Bdy%7D%7Bdt%7D)
Solving for dx/dt, we get
![\dfrac{dx}{dt} = -\left(\dfrac{y}{x}\right)\dfrac{dy}{dt} = -\left(\dfrac{\sqrt{L^2 - x^2}}{x}\right)\dfrac{dy}{dt}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdx%7D%7Bdt%7D%20%3D%20-%5Cleft%28%5Cdfrac%7By%7D%7Bx%7D%5Cright%29%5Cdfrac%7Bdy%7D%7Bdt%7D%20%3D%20-%5Cleft%28%5Cdfrac%7B%5Csqrt%7BL%5E2%20-%20x%5E2%7D%7D%7Bx%7D%5Cright%29%5Cdfrac%7Bdy%7D%7Bdt%7D)
We know that
![\dfrac{dy}{dt} = 4\:\text{ft/s}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdy%7D%7Bdt%7D%20%3D%204%5C%3A%5Ctext%7Bft%2Fs%7D)
when x = 6 ft. So the rate at which the top of the ladder is going down is
![\dfrac{dx}{dt} = -\left(\dfrac{\sqrt{100\:\text{ft}^2 - (6\:\text{ft})^2}}{6\:\text{ft}}\right)(4\:\text{ft/s})](https://tex.z-dn.net/?f=%5Cdfrac%7Bdx%7D%7Bdt%7D%20%3D%20-%5Cleft%28%5Cdfrac%7B%5Csqrt%7B100%5C%3A%5Ctext%7Bft%7D%5E2%20-%20%286%5C%3A%5Ctext%7Bft%7D%29%5E2%7D%7D%7B6%5C%3A%5Ctext%7Bft%7D%7D%5Cright%29%284%5C%3A%5Ctext%7Bft%2Fs%7D%29)
![\:\:\:\:\:\:\:= -5.3\:\text{ft/s}](https://tex.z-dn.net/?f=%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%20-5.3%5C%3A%5Ctext%7Bft%2Fs%7D)
The negative sign means that the distance x is decreasing as y is increasing.
Answer:
Positive correlation.
Step-by-step explanation:
The miles per hour increases as the cycling speed increases.
Answer:
254.34 cubed centimeters
Step-by-step explanation:
Answer:
b= -24/13
Step-by-step explanation: