MHB Help with O Level Math 4024 Exam Questions

[leylamew]

Hi, I'm leylamew.
So, I'm in the Cambridge University O Level board and I'm studying for my examinations, which will be in May. I'm taking the Math 4024 paper.
I was solving the past paper for 2010 and I hit a roadblock. (Shake)
Here's the link. http://papers.xtremepapers.com/CIE/Cambridge%20International%20O%20Level/Mathematics%20D%20(Calculator%20Version)%20(4024)/4024_s10_qp_21.pdf

It's questions 9bi, 10, 11 and 12 that are bothering me greatly.
Help? :D

 

[alyafey22]

Hi, I'm leylamew.
So, I'm in the Cambridge University O Level board and I'm studying for my examinations, which will be in May. I'm taking the Math 4024 paper.
I was solving the past paper for 2010 and I hit a roadblock. (Shake)
Here's the link. http://papers.xtremepapers.com/CIE/Cambridge%20International%20O%20Level/Mathematics%20D%20(Calculator%20Version)%20(4024)/4024_s10_qp_21.pdf

It's questions 9bi, 10, 11 and 12 that are bothering me greatly.
Help? :D

You are supposed to at least show some effort and tell us where are you stuck ?

 

[leylamew]

You are supposed to at least show some effort and tell us where are you stuck ?

I'd upload the sheet where I've scribbled the logic, but I don;t have a scanner.
Here's my thinking process:
OB=OA=OP=8 cm and as OQ is x cm, MN should be...something. But I can't understand for the life of me, how to find MN. I don't understand what I'm supposed to do with the x there.

- - - Updated - - -

I'd upload the sheet where I've scribbled the logic, but I don;t have a scanner.
Here's my thinking process:
OB=OA=OP=8 cm and as OQ is x cm, MN should be...something. But I can't understand for the life of me, how to find MN. I don't understand what I'm supposed to do with the x there.

And, I don't understand what the questions are asking of me in the others.

 

[leylamew]

I'd upload the sheet where I've scribbled the logic, but I don;t have a scanner.
Here's my thinking process:
OB=OA=OP=8 cm and as OQ is x cm, MN should be...something. But I can't understand for the life of me, how to find MN. I don't understand what I'm supposed to do with the x there.

- - - Updated - - -
And, I don't understand what the questions are asking of me in the others.

All right, so I put my ruler to use and hoping that the diagram was to scale, I measured MQ and OQ. They're the same, so I safely assumed that MQ is x and so, therefore, MN is 2x, as MQ is 1/2MN. Did I do it right?

 

[MarkFL]

All right, so I put my ruler to use and hoping that the diagram was to scale, I measured MQ and OQ. They're the same, so I safely assumed that MQ is x and so, therefore, MN is 2x, as MQ is 1/2MN. Did I do it right?

Yes, as triangles $MOQ$ and $NOQ$ are congruent, and are right isosceles triangles, we know that:

$$\overline{MQ}=x$$

$$\overline{QN}=x$$

and so adding, we have:

$$\overline{MQ}+\overline{QN}=2x$$

And we know that $$\overline{MQ}+\overline{QN}=\overline{MN}$$, hence:

$$\overline{MN}=2x$$

Now, for the next part of the problem, find the total area of the cross-section using the formula for the area of a circular sector (or simply from the fact that it is a quarter circle) and then subtract away the area of triangle $MON$. What do you find?

 

[topsquark]

$$\bar{MQ}=x$$

@MarkFL: Just a comment about the LaTeX coding: \overline{AB} gives \overline{AB}. I think that looks like a better operation here than \bar.

-Dan

 

[MarkFL]

Thank you, that does look better! (Happy)

 

[MarkFL]

...
Now, for the next part of the problem, find the total area of the cross-section using the formula for the area of a circular sector (or simply from the fact that it is a quarter circle) and then subtract away the area of triangle $MON$. What do you find?

Hints:

The area $A_S$ of a circular sector is:

$$A_S=\frac{1}{2}r^2\theta$$

and you know $r$ and $\theta$.

The area $A_T$ of a triangle is:

$$A_T=\frac{1}{2}bh$$

and you know the base $b$ and the height $h$ in terms of $x$. So you now need to compute:

$$A=A_S-A_T=?$$