Real math sat iic: how to do # 35, 48

Discus: SAT/ACT Tests and Test Preparation: June 2004 Archive: Real math sat iic: how to do # 35, 48
 By Teal (Teal) on Friday, June 04, 2004 - 01:19 am: Edit

I need the explanations to these questions..

35 (picture) Figure 8 shows a triangle inscribed in a semicircle. what is the area of the triangle in terms of theta? (the diameter is 2)
(((pg 221 REAl sat ii)

48. Which of the following has an element that is less than any other element in that set?
I. the set of positive rational numbers
II the set of positive rational numbers r such that r^2 >= 2
III. the set of positive rational numbers r such that r^2 >4

A. none
b. I only
c. ii only
d. iii only
e. i and iii

 By Aim78 (Aim78) on Friday, June 04, 2004 - 01:29 am: Edit

I don't have the book, so let me look at 48.

I is correct because 0 is smaller than any other number, so you can cross out a, c, and d.

II isn't correct because square root of 2 is not rational.

III...um...no, I don't think so. The smallest number is greater than 2, but by a TINY bit, it's like an asymptote.

 By Satchamp (Satchamp) on Friday, June 04, 2004 - 03:11 am: Edit

is 0 considered a positive Number.......Aim78??

 By Satchamp (Satchamp) on Friday, June 04, 2004 - 03:12 am: Edit

i would say both II and III look right to me....but thats not an option.so im definitely retarded.....oh well
wuts the answer in the book??

 By Joshjmgs (Joshjmgs) on Friday, June 04, 2004 - 06:56 am: Edit

48 was A (none) but i dont know WHY

35.
Area of a tyriangle euals (.5)(base)(height)
Now the base of the triangle equals 2sinx (just basic trig)
and the height equals 2cosx
sow e have
(1/2)(2cos)(2sin)
simplifies to
2sin(x)cos(x)
which simplifies to
sin(2x)

 By Joshjmgs (Joshjmgs) on Friday, June 04, 2004 - 07:01 am: Edit

OK i got the answer for 48
48. Which of the following has an element that is less than any other element in that set?
I. the set of positive rational numbers
II the set of positive rational numbers r such that r^2 >= 2
III. the set of positive rational numbers r such that r^2 >4

I the set is all intergers greater than 1
II All intergers greater than 2 (root 2 is irrational)
III Since it cannot equal 4, it must be greater than 2 (same as II)

So it MIGHT be that they all have a lower bound for their sets (?)

 By Ecnerwalc (Ecnerwalc) on Friday, June 04, 2004 - 08:34 am: Edit

0 is not positive.

 By Optimizerdad (Optimizerdad) on Friday, June 04, 2004 - 09:26 am: Edit

48. Consider the related question 'Does this set of rational numbers have an element that is less than any other element in that set? '

I. The set of positive rational numbers? No. 0 does not belong to this set. If you give me any rational number x/y and claim that it is the smallest number in this set, I can *always* find one that is smaller e.g. x/2y

II. The set of positive rational numbers r such that r^2 >= 2 ? No. Sqrt(2) is irrational, as was pointed out by Joshjmgs. Again, if you give me any rational number x/y from this set, and it has n significant digits, I can always evaluate Sqrt(2) to an accuracy of (n+1) digits, and this will be closer to the true value of Sqrt(2) than x/y .

III. The set of positive rational numbers r such that r^2 >4? No. The logic is similar to that used for I; if you claim that x/y is the smallest number in this set, I can always generate a new number x/y - 0.5*(x/y -2) that is closer to 2.

 By 1212 (1212) on Friday, June 04, 2004 - 10:33 am: Edit

yep optimizerdad is right, you can find a smaller positive rational number for all 3 choices at any time, therefore none.

 By Madd87 (Madd87) on Friday, June 04, 2004 - 12:00 pm: Edit

I got none as well, all I did was look at it and say there is always a number smaller than any number I pick because its not "or equal to" (except II but that was irrational).

ie. .00001 is close to 0 and is rational and positive, but .000001 is closer, and .0000001 is closer than that.