The top of a 20-foot ladder leans against a wall, and the bottom of the ladder slides away from the wall at a rate of 5 ft/sec. find the velocity of the top of the ladder at time t=1 sec if the bottom of the ladder is 7 feet away from the wall at time t=0

vy= -3.75 ft/sec (slides down )

Explanation:

denoting y as the coordinate of the top of the ladder , x as the coordinate of the bottom , L as the length of the ladder , then

x²+y²=L² = constant

deriving by t

2*x*dx/dt + 2*y*dy/dt = 0

since the velocity is vx=dx/dt and vy=dy/dt

x*vx+y*vy=0

then since the velocity of x is constant x=x₀+vx*t

at t=1 sec x=7 ft +  5 ft/sec*1 sec = 12 ft

then x²+y²=L² → y= √(L²-x²) =√[(20 ft)²-(12 ft)²] = 16 ft

after that

x*vx+y*vy=0 → vy=  -(x/y)*vx = -(12 ft/16 ft)*5 ft/sec= -3.75 ft/sec

vy= -3.75 ft/sec (slides down since vx is positive, that means that slides away)

Note

we cannot assume that vy is constant ( and do y=y₀+vy*t)  when vx is constant , since y= √(L²-x²) = √(L²-(x₀+vx*t)²) , that is not linear with t . Nevertheless, vy could be found deriving y with respect to t , but that would be more complicated than the other way.

the working equation for the angular size is the product of the physical size and 360 degrees divided by the product of two pi and distance from earth. the minimum size would be 0.559 degrees while the maximum size would be 0.49 degrees. 

a cold ice-tea.

pour in room temperature or cold water. cover the pitcher and refrigerate. steep white or green tea for 6 to 8 hours; steep black or oolong tea for 8 to 12 hours.jun 28, 2015