If 3^x = k, what does 3^(2x+1) equal in terms of k?
(A) 3k^2
(B) 9k^2
© k^2 + 3
(D) 2k + 3
(E) 2k + 1
Could you also explain what type of problem this is?
It is © k^2+1. This is a common algebraic-manipulation type of problem. Are you familiar with all the properties of numbers with exponents? Let me know if you just want the solution.
@JuicyMango it is (A):
3^(2x+1) = 3^(2x) * 3^1 = 3*(3^x)^2 = 3k^2.
Can also be solved by plugging in numbers, e.g. x = 2 and k = 9.
@MITer94: Oh woops thanks for catching that; in retrospect I should not have tried to do the problem in my head :P.
But regardless, you should have a familiar understanding of exponents and their properties with this type of question.
Ah. I always understand the arithmetic involved, but I just don’t see what I should do to start… I should’ve just tried to simplify. Thank you for the responses.
I have looked up some solutions to this problem, but I just cannot picture the solution for some reason. I know that it some sort of transformation is required to change the figure into one that is a rectangle, but I just cannot imagine it. http://prntscr.com/7tlafb
After thinking about this question for some time each day, it finally dawned on me how to get to the solution (yes, like 10 minutes after posting the question). I guess I’ll answer my own question, if anyone else was curious.
http://prntscr.com/7tldz2
Here’s a tricky one (especially if you’re doing it algebraically):
If |a-b| = 5, and |a-c| = 3, which of the following could be the value of |b-c| ?
I. 2
II. 4
III. 8
(A) I only
(B) II only
© III only
(D) I and II
(E) I and III
This question looks flawed. Using the triangle rule any of those 3 should be possible. But that’s not an answer choice!
Eh, excuse me @DrSteve , but how can 4 be possible?
Whoops - I read the question too quickly (and I think I was tricked by the fact that the poster said the question is tricky): 4 isn’t possible. The answer is in fact (E).
It’s easy to see that there are only 2 possibilities geometrically by displaying a, b, and c on the real line in the orders a, c, b and c, a, b
To solve this algebraically (not recommended) would require 4 cases.
This is not a triangle rule problem since a, b, and c are real numbers and not points.
A few technical things I want to mention that may help some of you with these types of problems:
(1) Geometrically, |a - b| is “the distance between a and b.”
(2) |a - b| = |b - a|
In other words, “the distance between a and b” is the same as “the distance between b and a.”
(3) Algebraically |a - b| = 5 is equivalent to two equations without absolute value: a - b = 5, a - b = -5
Alright I cant figure this one out without taking more than 5 minutes.
In the xy-coordinate plane, what is the area of the triangle whose vertices are the points with the coordinates (3,2) (5,1) and (8,4)
Possible Answers are
A - 4
B - 4.5
C - 5
D - 5.5
E - 6
Oh wait got it… (5,1) to (8,4) had the distance of 3, and the height is 3, making it 3 x 3 = 9, 9/2 =4.5
@chyaboi123 nope – distance between (5,1) and (8,4) is 3 sqrt(2). The area is still 4.5 though. Several ways to get the correct answer, but if you know the shoelace formula, it gives a very quick solution:
3 2
5 1
8 4
3 2
Area = (1/2)|31 + 54 + 82 - 52 - 81 - 34| = (1/2)9 = 4.5
@MITer94 Could you explain the shoelace formula?
@chyaboi123 in general, the formula goes like this:
Given an n-gon with vertices in the plane (x1, y1), (x2, y2), … (xn, yn) in order (counterclockwise or clockwise), line up the vertices in an (n+1)*2 matrix starting and ending with the first vertex:
x1 y1
x2 y2
x3 y3
…
xn yn
x1 y1
Now take the two sums
S = x1y2 + x2y3 + … + xny1
T = x2y1 + x3y2 + … + x1yn
(it will look like the laces of a shoe if you draw diagonal lines representing the addends)
Then the area of the polygon is (1/2)*|S-T|.
The Wikipedia article gives a couple examples.
You won’t need the shoelace formula for the SAT, but it’s a pretty neat theorem anyway.
E
@chyaboi123 Or, you could add a program to your calculator that finds it for you. Check this out : (http://www.ticalc.org/pub/83plus/basic/math/geometry/)