Math Question need help with....

[umer007]

There is a semi-circle with diameter XY. Rectange PQRS is inscribed in the semi-circle with PQ=12 and QR=28. Square STUV has T on RS, U on the semi-circle and V on XY. The area of STUV is closest to....

 

Answer is 16.

 

I just want to know steps to come up to this answer. I made a drawing on paint, it dznt look xactly like the one I have but it shd give u guyz and idea of the question. If someone can show me steps to get to this answer it wd b helpful alot.

 

Thx.

[albertlee]

Not closest to 16, but it is exactly 16!!!

 

The solution is:

 

1) Call the center of XY, O

 

2) We know that QP = RS, hence the angle, QOP = ROS, therefore, PO = OS

 

3) from 2), we know the radius is QO or RO, and we also know that PS = 28, and RS = 12, so according to Pytha. theorem, RO = root(144+196) = root(340) = radius

 

4) now parameterize TU = x, which equals to SV, we know that OS = 14, ie, half of PS, so OV = 14 + x

 

5) we know the radius is root(340), using pytha. theorem again, that 340 = x² + (x+14)²

 

6) now rearange 5), we get the quadratic, x² + 14x - 72, and its factorized form will be (x+18)(x-4)

 

7) x could either be -18 or 4, -18 is not possible, because we cant have a side of square being negative, ie, it is not a vector quantity, so 4 is the answer for x

 

8) Finally....... 4*4 = 16!!!!!

 

 

Does that help???

 

Albert

[timo]

Well, I can tell you the steps I´d take to get the answer:

a) To deternime the diameter of the circle you use the height-sentence (or whatever it´s called in english): (PQ)² = XP * PY = XP * (PS + SY) = XP * (QR + XP).

Solving for XP gives you the position of all points except T, U and S

b) To determine VU and thus the position of the other points you can use the same method again: (UV)² = (XS + SV) * (XY - XS - SV)

Note that UV = SV because SVUT is a square. Solve and be happy

 

 

EDIT: Hehe, 12 hours without response and then two people answering within 10 minutes .

[umer007]

oic, thx alot. I understand how to get it from the first response but the 2nd response confuses me Maybe if u go a bit slower, but thats alrite atleast I know one way now. Thanks again both of u.

 

Well another answer another question:P

 

A solid cube of side length 4 cm is cut into two peices by a plane that passed through the midpoints of six edges, as shown. To the nearest centrimetre, the surface area of each half cube created is?

 

I have learned the answer to be 69....is this right bcuz i havent learnd how to prove it:P

 

personally i get confused by just looking at the picture, where are these half cubes??? are they supposed to be infront and behind the hexagon?

Doc1.doc

[albertlee]

Ok, for that question, first of all, find the area of the hexagon....

 

first find the length of 1 side of hexagon, which is root(8) or 2*root(2)

 

the area of the hexagon will be [root(2)*root(6)]*6, which is = root(12)*6 or 2*root(3)*6 which is 12*root(3)

 

 

Can you follow the above?

 

 

Then, Think about how many sides there will be, 7, right??? the cube has 6 sides, plus a side of a plane cutting through, you can "visualize" in my picture

 

 

now you have already known the area of hexagon, so 6 sides left

 

Let's start with, what's the area of 1 red triangle???? area of 1 red triangle is 2

 

and how many red triangles are there??? 3, so the area of 3 of them is 6

 

 

now we have 3 sides left, the other 3 sides are all equaled, which 1 of them has an area of square subtracted by the blue triangle, which is same area of the red triangle, so, the side of the cube is 4*4=16, 3 of them is 48, subtracting 3 triangles so is 48-6 = 42

 

now the total surface area is the total of numbers in bold, ie, the answer is: [12*root(3)]+6+42, which should be approximately 69, I guess.

 

Does that help??

 

Albert