Differentiation-rate of change

2009-01-11 8:34 am

1. A ball is thrown upwards. Its height s metres from the ground after t seconds is s = 30t - 5t^2.
(a) What is the velocity of the ball at time t?
(b) At what time does the ball reaches its highest point?
(c) What is the velocity of the ball just before hitting the ground?

2. A person modifies the size of a rectangular frame displayed on the monitor of a computer by dragging a mouse. The current size of the frame is 15 cm by 12 cm.
(a) If the length of the frame increases at a rate 0.8 cm/s and the width increases at 0.5 cm/s, find the rate of increase of the perimeter and the area of the frame.
(b) If the width decreases at a rate 2 cm/s while the area of the frame remains a constant, what is the rate of change of the length?

回答 (1)

Ray

2009-01-11 9:42 am

✔ 最佳答案

1.(a)
v= s' = 30 - 10t

1.(b)
s = 30t - 5t^2
s = -5( t^2 - 6t + 9) + 45
s = -5 (t - 3)^2 + 45
When t = 3, s is at its maximum (45)

1.(c)
When s= 0,
-5t^2 + 30t = 0
t^2 - 6t = 0
t(t - 6)=0
t= 0 or t=6

v = 30 - 10(6) = -30

2.(a)
Let l, w, p, A be the length, width, perimeter, area of the frame respectively.

p = 2(l + w)
p' = 2( l' + w')
p' = 2(0.8 + 0.5) = 2.6 cm/s

A = lw
A' = lw' + wl'
A' = 15(0.5) + 12(0.8) = 17.1 cm^2/s

2(b)
A' = lw' + wl'
0 = 15(-2) + 12l'
l' = 2.5 cm/s

參考: Myself

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