Pure math(p3)

[astarmathsandphysics]

can someone solve the last part please
An underground storage tank is being filled with liquid as shown in the diagram. Initially the tank is
empty. At time t hours after filling begins, the volume of liquid is V m3 and the depth of liquid is h m.
It is given that V = 4
3h3.
The liquid is poured in at a rate of 20m3 per hour, but owing to leakage, liquid is lost at a rate
proportional to h2. When h = 1,
dh
dt = 4.95.
(i) Show that h satisfies the differential equation
dh
dt = 5
h2 ? 1
20
. [4]
(ii) Verify that
20h2
100 ?h2 ? ?20 + 2000
(10 ?h)(10 +h). [1]
(iii) Hence solve the differential equation in part (i), obtaining an expression for t in terms of h.

in this thread

https://studentforums.biz/index.php/topic,740.0.html

[ITS MEE...]

thank u

[astarmathsandphysics]

The polynomial x^4 - 2x^3 - 2x^2 +a is denoted by f(x). It is given that f(x) is divisible by x^2 - 4x +4.
(i) Find the value of a.
(ii) When a has this value, show that f(x) is never negative.


can someone please explain part (ii)
i don't understand the question

 x^2 - 4x +4.=(x-2)^2 so put x=2 and solve f(x)=0 for a

16-16-8+a=a=0 to geta=8 Now do long division by  x^2 - 4x +4.to getx^2+2x+2=(x+1)^2+1>0 and since(x-2)^2 is never -ve neither is the product of it and (x+1)^2+1

[ITS MEE...]

can u integrate
sin^3xcosx
please

[ITS MEE...]

oh i got it no need to solve it but can u integrate this
(1+tanx)^4 sec^2x

[twilight]

can u integrate
sin^3xcosx
please

how didya solv it?

[ITS MEE...]

well wnever u see an ODD one ie unfamiliar and kind of compound just try to differentiate one of them and if u get the other (regardless of the power or something like a number multiplied or divided by it then u intergrate the one u differentiated
  hope its a bit clear
Astarmathsandphysics can u explain it a bit more or make it less vague? please

[louis]

ii) Find a vector equation for the line of intersection between two planes :
      2x -y - 3z= 7 and x + 2y + 2z=0     
Solution :  The normal vectors of 2 planes are n1= ( 2,-1,-3)    and n2=  (1,2,2)
                    These 2 vectors form a new plane.
                    The normal vector of this new plane is  ( 2,-1,-3)  X   (1,2,2)  =  (4, -7, 5)
                    The normal vector is parallel to the line of intersection of the 2 planes.
                    So the direction vector for the line of intersection of 2 planes is  a ( 4, -7, 5 )

                    Now we have to find the position vector of a point on the line of intersection
                    In the equations of  2x - y -3z= 7 and x + 2y + 2z = 0,there are 3 unknowns.
                    Reduce to 2 equations with 2 unknowns only by putting x=0  or  y=0  or z=o
                    We  choose y=0
                    Solve for x and z in the equations  2x -3Z = 7
                                                                    x +  2Z = 0
                                                     we get        Z   =  -1     x =  2
                     So the vector  equation for the line of intersection of 2 planes is
                                                     r=  2i -k  + a(  4,-7 ,  5 )

[twilight]

ooo i think i get wut iu mean ITS ME..
so is it    (1/4)(sin⁴x) ?

[astarmathsandphysics]

oh i got it no need to solve it but can u integrate this
(1+tanx)^4 sec^2x

Substitute u=1+tanx to get u^4dv ans=1/5(1+tanx)^5

[ITS MEE...]

THANK U SOOOOOOOOOO MUCH LOUIS

[astarmathsandphysics]

can u integrate
sin^3xcosx
please

how didya solv it?

Substitute u=1+tanx to get u^4du ans=1/5(1+tanx)^5

[twilight]

can u integrate
sin^3xcosx
please

how didya solv it?

Substitute u=1+tanx to get u^4du ans=1/5(1+tanx)^5

ummmm .. wher is the tan???

[ITS MEE...]

THANK U Astarmathsandphysics AND TWIGHLIGHT IT RIGHT

[twilight]

thanks evry1