MT 1Chapter differentiation 的问题

ting_006

Rate of change question:

1.Sand is poured onto a horizontal floor at a rate of 30pie cm^3s^-1 and form a pile in the shape of a right circular cone,the height of which os half the radius.Calculate the rate of change of the radius when is 10cm.

Ans:0.6cms^-1


2.A ladder 5m long against a vertical wall.The bottom of the ladder is pulled along the ground away from the wall at a constant rate 0.3ms^-1.How fast will the top of the ladder be falling at the instant when is bottom is 2m away from the wall?

Ans:0.13ms^-1


3.A hemispherical bowl of radius 8cm contains water which is flowing into it at a constant rate.When the height of the water is h cm,the volume  Vcm^3 of the water in the bowl is given by

                V=pie(8h-1/3h^3)

Find the rate at which the water level is rising when h=5 ,given that the time taken to fill the bowl is 2minutes.

Ans:0.052cms^-1



Problems concerning Maximum and minimum values question:

1.If V=x^2y and x+y =12 where x and y are positive ,find the maximim value of V.

Ans:256


2.Find the minimum value of x^2 +xy +y^2 if x and y are connected by the relation 2x+y=1.
Ans:1/4

3.A piece of wire of the length 16cm is cut into two pieces,one of the length x cm and the other of leng (16-x) cm.The piece of the length x cm is bent to form a circle with circumference x cm.The other piece is bent to form a square with perimeter (16-x) cm .Show that as x varies,the sum of the areas enclosed by these two pieces of wire is a minimum when the radius of the circle is 8/4+pie cm.


4.A rectangle is inscribed in a semicircle with a fixed radius R as shown. Two vertices of the rectangle lie on the arc of the semicircle while the other two lie on the diameter of the semicircle.
Find the greatest area of the rectangle that can be inscribed.

Ans:R^2

这几题问题我真的算不到，请各位高手帮帮忙，麻烦了！

elainehappy

我想问，account的话，form6有需要MT嘛???

peaceboy

回复 1# ting_006


    太多题的pasal我恨懒惰一次过做完


1.Sand is poured onto a horizontal floor at a rate of 30pie cm^3s^-1 and form a pile in the shape of a right circular cone,the height of which os half the radius.Calculate the rate of change of the radius when is 10cm.

Ans:0.6cms^-1

dv/ds = 30 pi
v=(1/3) pi r^2 h
h= (1/2) r
v = (1/6) pi r^3
dv/dr =  (1/2)pi r^2
dr/dv = 2/( pi r^2)
dr/ds =  dr/dv * dv/ds
         =2/(pi r^2) * 30 pi
          =60/r^2
r=10
dr/ds = 0.6

Log

回复 3# peaceboy
不要这么懒吧。。。

peaceboy

回复 4# Log

2.A ladder 5m long against a vertical wall.The bottom of the ladder is pulled along the ground away from the wall at a constant rate 0.3ms^-1.How fast will the top of the ladder be falling at the instant when is bottom is 2m away from the wall?


dw/ds = 0.3

h^2 = 5^2 - w^2
h = (5^2 - w^2)^(1/2)

dh/dw = [-2w]/[2(5^2 - w^2)^(1/2)]
          = -w/(5^2 - w^2)^(1/2)

dh/ds = dh/dw * dw/ds
          = 0.3 * -w/(5^2 - w^2)^(1/2)
          = -0.3w/(5^2 - w^2)^(1/2)
w=2
dh/ds=0.13






    剩下的你来做

ting_006

谢谢帮忙!!!!
感激!!!感激!!!!!!!!