iamdan08's spec questions

[evaporade]

N = 3000e^(1/3ln(3)t) - 4500(t - 3), for t > 3.

[iamdan08]

Thanks. But the next question asks how long does it take for the bacteria to die out...

[d0minicz]

that would mean you would have to find t when P = 0

[iamdan08]

Yeah i know. But when i solve it on my calculator i don't get a solution...

[evaporade]

To die out means less than 1.

It occurs when t > 3.0020235, i.e. 7.3 seconds after the wash.

[iamdan08]

Oh ok that makes sense! Thanks!

[iamdan08]

Ok here are a few vector questions:               

1) find the point which divides the line segment containing the point A with position vector (1,2,3) and the point B with position vector (-1,4,2) in the ratio 2:3.

2) let ABC be a triangle and let D divide the line segment BC in the ratio of 3:5. Express the vector AD as a combinant of AB and AC.

3) a trapezium is a quadrilateral with 2 parallel sides. Show that the points A(4,3,0), B(5,2,3), C(4,-1,3) and D(2,1,-3) form a trapezium.
How would I show this?

Thanks

[Mao]

1) A + 2/5 AB

2) AB + 3/8 BC

3) AB = k*CD (and AC k*BD)

[iamdan08]

Thanks. But is there any way of explaining them?

[kamil9876]

.... there is:

1.) There exists a line AB. Split the line into 5 parts, point C is 2 parts away from A and 3 parts away from B. Hence
However

(cordinates of A) 

2.)

AD=AB+BD

Again do a similair thing to BD, split it into 8 parts, and we know that D is 3 parts away from B.

3.) Two lines must be parralel, parralel lines have the same gradient:

consider the vectors:

xi+yj
ai+bj

If they are parrelel, rise over run is the same hence:

y/x=b/a
y/b=x/a

let y/b=x/a=k

therefore:

yi+xj=k(ai+bj)

So in order for vectors to be parrelel, they must be multiples of each other (other than k=0). Hence you must find two vectors that are parrelel(multiples of) with each other from the possibilities AB,AC,AD,BC,BD,CD

[iamdan08]

Awesome thanks! Appreciate it!!! 

[iamdan08]

Ok, i have saved up  a few more questions  : :

Q1) Find dy/dx by implicit differentiation:

=

Q2) A point P=P(x,y) moves along the line y=3x, while point Q is stationary at (2,0).  Write down an expression for the distance D between ponts P and Q, and express this soley in terms of x. How fast is the distance between P and Q changing at the instant when P(x,y)=P(2,6), if x is decreasing at a rate of 5m/s at that instant?

Q3) Water leaks from an inverted conical tank at a rate of 10 000cm^3/min. At the same time water from another source is entering the tank at a constant rate Q. The tank is 6m high and the diameter at the top is 4m. If the water level is rising at a rate of 20cm/min when the height of the water is 2m, find Q.

Q4) A ladder 10m long rests agaisnt a verticle wall. If the bottom of the ladder slides away from the wall at a speed 2m/s, how fast is the angle between the top of the ladder and the wall changing when the angle is

Q5) A kite flies at height:

  0t10.
a) What is the maximum height reached by the kite in the fist 5 mins?
b) What is the maximum height reached by the kite in the whole 10 minutes?

Q6) FInd the maximum possible area of a rectangle which has its base on the x-axis and its upper verticies on the semi-circle , 3x3.

Thanks Heaps!!!!!

[kamil9876]

Q1:



Differentiate both sides, product rule on left chain rule on right:



Now I'm only going to focus on the first term on the left, I think ther other two you should handle ok:
Just keep using the chain rule until you only get and no other derivatives:





The main idea in implicit differentiation is to just keep differentiating until you get only and no other derivative expressions. Then just collect and factorise all the to get it by itself

[kamil9876]

2.)

We can gather from just the first sentence that the co-ordinates of P are governed by:


(justify this by drawing a diagram. Distance between two points is just an application of the pythagorean theorem to cartesian co-ordinates (Break the line up into horizontal and vertical components))

Now the x and the y respresent co-ordinates of P, however we know that y=3x for a point that can be P, hence sub that in to get it in terms of x only.

The rate of x with respect to t:

Find

We are after hence chain rule would help as we know/can easily find out the derivatives on the right:

[kamil9876]

Q4.)
is the angle between top of ladder and wall.







Now use chain rule to find