Differentiation

用戶225241

2008-10-13 6:55 am

1. Suppose the distance s between a moving point P and a fixed piont Q after
a certain time t is given by s= t^3 -9t^2 +24t where s is measured in meters
and t in seconds
a. Find the velocity and the acceleration when t =5
b. Find the value of t when P is momentarily at rest

2. A ladder of 5m leans against wall. If the lower end of the ladder slips at the
rate of 4m/sec and is 3m from the wall, how fast is the upper end of the
ladder comming down the wall at that moment

回答 (1)

momo

2008-10-13 8:08 am

✔ 最佳答案

(1) (a) s = t^3 - 9t^2 + 24t
v = ds/dt = 3t^2 - 18t + 24
a = d/dt(ds/dt) = 6t - 18
when t = 5, v = 3(5)^2 - 18(5) + 24 = 9m/s
when t = 5, a = 6(5) - 18 = 12m/s^2
(b) when P is at rest, ds/dt = 0
3t^2 - 18t + 24 = 0
t^2 - 6t + 4 = 0
t = [6 √(6^2 - 4(1)(4))]/2 = 3√5 = 0.764s or 5.236s
(2)
Let the horizontal distance of the lower end of the ladder to wall be x
and vertical distance of upper end of ladder to ground be y
5^2 = x^2 + y^2
y = √( 25 - x^2)
differentiate the equation w.r.t.t
dy/dt = [0.5(-2x)/√(25 - x^2)](dx/dt)
= [-x /√(x^2+25)](dx/dt)
when x = 3m, dx/dt = 4m/s
dy/dt = [-3/√(-3^2 + 25)] (4) = -3m/s
rate of change of the vertical distance of the ladder = 3m/s towards ground

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