Need 3u Yr11 Cambridge Question Solutions (1 Viewer)

[Tsylana]

Does anyone have the solutions for exercise 8E from questions 23 onward ><"

 

[lyounamu]

Re: Need Cambridge Question Solutions

Does anyone have the solutions for exercise 8E from questions 23 onward ><"

Which textbook is it? 2 Unit or 3 Unit? What about the year?

 

[Tsylana]

Re: Need Cambridge Question Solutions

3U, Year 11, ahh sorry, the quadratic ones... ><"

 

[lyounamu]

23. Let l = length of a string
And l = perimeter of a sector
Therefore, l = r@ (i.e. arc legnth) + 2r (i.e two radii)
Manipulating this, you get:
r@ = l - 2r
@ = (l-2r)/r

Now, the area of a sector = 1/2 . r^2 . @
= 1/2 . r^2 . (l-2r)/r (I substitued the value of @)
= 1/2(rl - 2r^2)
= 1/2rl - r^2
Now, dA/dr = 1/2 l - 2r

The maximum area occurs at dA/dx = 0.
i.e. when 1/2 l - 2r = 0

Therefore, r = 1/4 l

I am sorry. I could not be bothered to prove that it is the maximum point but it is very obvious that this is the maximum point. However, when you get this question, you ALWAYS need to prove. Let me try Q 24 now.

 

[Tsylana]

Hmm, i've never been great at geometry... but how is the arc length related to the radius? "

 

[lyounamu]

Hmm, i've never been great at geometry... but how is the arc length related to the radius? "

You are only at the Preliminary level of Mathematics. So you won't know it.

If you go to HSC Mathematics, you learn that arc length = r@ where @ is a degree in radians.

It seems to me that all the quesitions you posted are quite redundant in that case since you have not learnt them.

I actually tried doing Q24, but I cannot seem to be able to get it. Solution for that would be a great help.

 

[lolokay]

I am sorry. I could not be bothered to prove that it is the maximum point but it is very obvious that this is the maximum point. However, when you get this question, you ALWAYS need to prove.

it probably would have taken less time to prove it than to write that

You are only at the Preliminary level of Mathematics. So you won't know it.

If you go to HSC Mathematics, you learn that arc length = r@ where @ is a degree in radians.

It seems to me that all the quesitions you posted are quite redundant in that case since you have not learnt them.

I actually tried doing Q24, but I cannot seem to be able to get it. Solution for that would be a great help.

shouldn't you be able to easily figure out the relationship between the arc length and r? since you know the arc is @/2pi * 2pi r, or don't you learn radians in year 11?

what's question 24?

 

[Tsylana]

heh... ><" oh wells, i actually figured it out by deriving it through a bit of algebra and trig. With using your pi2r as diameter for a circle as a whole and cutting it down to a specific sector. So I could well say the question was not completely out of my reach x___X.

^ though i have no idea what the guy above me is saying.

Question 24 : Prove that the rectangle is the greatest area that can be incscribed in a circle is a square, Hint : recall that the maximum of A occurs when the maximum of A^2 occurs.

Question 25: The sum of the radii of two circles remains constant. Prove that the sum of the area of the circles is least when the circles are congruent.

meh... the other questions... just seem a bit out of my reach so i won't even bother posting them up.

 

[lyounamu]

it probably would have taken less time to prove it than to write that

Lol, that's true.

 

[lyounamu]

what's question 24?

Where you have to prove why the biggest rectangle in the circle is a square. It is a common sense question. But unfortunately, it is a mathematics question as well. I cannot use my superior english abilities over this (joking).

 

[Tsylana]

Lol. I dont see how the common sense works behind that question, but like if u just look at it is right... but like if someone told u to find the biggest 4 sided figure in that could fit into a circle i'd probably hesitate before i said a square.. O_O"

 

[lyounamu]

I just used circumference of a circle.

How?

 

[lolokay]

Lol. I dont see how the common sense works behind that question, but like if u just look at it is right... but like if someone told u to find the biggest 4 sided figure in that could fit into a circle i'd probably hesitate before i said a square.. O_O"

it says biggest rectangle, not any quadrilateral so you shouldn't need to hesitate before giving that answer

*

let the sides of the rectangle be x and y
let d = diagonal of rectangle = diameter of circle (therefore a constant) /would proof be needed that the diagonal of the rectangle is equal to the diameter of the circle?
so x^2 + y^2 = d^2
y = (d^2 - x^2)^1/2
area, A, of the rectangle = xy
= x(d^2 - x^2)^1/2
let d^2 - x^2 = u
dA/du*du/dx = 1/2u^-1/2 * -2x = x(d^2 - x^2)^-1/2
A' = 1(d^2 - x^2)^1/2 - x^2(d^2 - x^2)^-1/2 = 0 at max
d^2 - x^2 - x^2 = 0
d^2 = 2x^2
y^2 + x^2 = 2x^2
y^2 = 2x^2

when dA/dx is increasing
d^2 - x^2 - x^2 > 0
2x^2 < x^2 + y^2
x^2 < y^2
so this is a max, as area only increases as x approaches y

 

[Aplus]

How?

Wrong question.