contestada

(1 point) Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley P. The point Q is on the floor 12ft directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2ft/s. How fast is cart B moving toward Q at the instant when cart A is 5ft from Q

Respuesta :

Answer:[tex]\frac{20}{23} ft/s[/tex]

Step-by-step explanation:

Given

Velocity of point A is 2 ft/s

Length of rope is 39 ft

let x be the distance of A from Q and y be the distance of point B from Q

thus From Diagram

[tex]x^2+12^2=L^2[/tex]

Differentiating

[tex]2x\frac{\mathrm{d} x}{\mathrm{d} t}+0=2L\cdot \frac{\mathrm{d} L}{\mathrm{d} t}[/tex]

[tex]x\frac{\mathrm{d} x}{\mathrm{d} t}=L\cdot \frac{\mathrm{d} L}{\mathrm{d} t}[/tex]

at x=5 ft L=13 ft

thus [tex]\frac{\mathrm{d} L}{\mathrm{d} t}=\frac{5\times 2}{13}=\frac{10}{13}[/tex]

For point B

[tex]y^2+12^2=(39-L)^2[/tex]

Differentiate w.r.t time

[tex]2y\frac{\mathrm{d} y}{\mathrm{d} t}+0=2L\cdot \frac{\mathrm{d} 39-L}{\mathrm{d} t}[/tex]

[tex]y\frac{\mathrm{d} y}{\mathrm{d} t}=-(39-L)\cdot \frac{\mathrm{d} (39-L)}{\mathrm{d} t}[/tex]

at [tex]x=5\ ft[/tex]

[tex]y=23 ft[/tex]

thus [tex]\frac{\mathrm{d} y}{\mathrm{d} t}=\frac{-20}{23}ft/s[/tex]

Ver imagen nuuk
ankitprmr2

Answer:

[tex]\rm Therefore,velocity\; of\;B = \dfrac {20}{23}[/tex]

Step-by-step explanation:

Given :

Velocity of A = 2ft/s

Rope length = 39 ft

Let x be the distance between A and Q (refer the image).

Let y be the distance between Q and B (refer the image).

Thus from the image,

[tex]x^2 + 12^2 = L^2[/tex]

Now differentiating the above equation with respect to time t

[tex]2x \frac {dx}{dt} +0 = 2L\frac{dL}{dt}[/tex]

at [tex]\rm x = 5 ft[/tex] and  [tex]\rm L = 13ft[/tex]

[tex]\dfrac {dL}{dt}= \dfrac{5\times2}{13} = \dfrac{10}{13}[/tex]

For B point,

[tex]y^2 +12^2 = (39-L)^2[/tex]         (refer the image)

Now differentiating,

[tex]2y\frac{dy}{dt} + 0 = -2(39-L)\frac{dL}{dt}[/tex]

So, at [tex]\rm x=5ft[/tex] and [tex]\rm y = 23ft[/tex]

[tex]\dfrac {dy}{dt} = \dfrac{-20}{23}[/tex]

(negative sign represents opposite direction)

[tex]\rm Therefore,velocity\; of\; B = \dfrac {20}{23}[/tex]

For more information, refer the link given below

https://brainly.com/question/8127621?referrer=searchResults

Ver imagen ankitprmr2

Otras preguntas