1)For all positive angles less than 360 degrees, if csc(2x + 30degrees) = cos(3y-15degrees) the sum of x and y is
a) 30 degrees
b) 65 degrees
c) 95 degrees
d) 185 degrees
e) 215 degrees
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Year population(in million)
1950 25.8
1960 34.9
1970 48.2
1980 66.8
1990 81.1
the table shows the population of mexico at the beginning of five decades. Assuming the population has grown exponential, what population would have been predicted for 2000?
a)94.1
b)102
c)114.1
d)122.6
e)128.3
@gameplayer1234 I am a math major and am struggling to come up with an answer to 1 – very likely 1. is a bad problem, or you mis-typed it. Where is the problem from?
Use the definition csc x = 1/sin x.
1/sin (2x+30°) = cos (3y-15°) <==> sin (2x+30°) cos (3y-15°) = 1. Two cases:
Case 1: sin (2x+30°) = cos (3y-15°) = 1.
Then 2x+30° = 90°, 450°, etc. which implies x = 30°, 210°, 330° (I am assuming 0° < x,y < 360° by the constraints of the problem)
Also 3y-15° = 0°, 360°, 720°, etc. and the only solutions y in the range are y = 5°, 125°, 245°, etc. So it becomes clear that the value of x+y is not well-defined. Taking the smallest solutions (x = 30°, y = 5°) results in a true statement but is not in the answer choices.
Case 2: sin (2x+30°) = cos (3y-15°) = -1 produces x = 120°, 300° and y = 65°, 185°, etc.
However if it is constrained that all trig arguments have to be strictly between 0° and 360° then there is a unique answer (x = 120°, y = 65°) leading to an answer of D. But this question is worded extremely poorly.
- You should use exponential regression (most, if not all, graphing calculators have this). Plugging the data into an exponential regression calculator, you should get something like:
P(year) (in millions) = 3.3283*10^(-24) * (1.02983)^(year)
According to the regression equation, P(2000) ≈ 114.1 (million), so the best answer is C.
Also, are you sure you have the data right for #2? If the growth is exponential, that last data point looks kinda low.
The reason I am asking is that in exponential growth, the function increases by (roughly) equal percents in equal time intervals. The first changes are about 35%, then 38% then 38% but then only about 20%.
It’s looking to me like one thing hasn’t changed with the new SAT: it’s still easy to get distracted by poorly written non-College Board questions…
@pckeller Also I just noticed, the population increase between 1980 and 1990 is less than the population increase from 1970 - 1980. However, doing an exponential regression on that data yielded an estimate that matched an answer choice…
I still believe that #1 is poorly written.
Yeah, #1 is not clear. But I’d say that if the angles [What angles?] are positive but less than 360, then I agree that the only solution is x=120, y= 65…
^Or if it specifically said that 0 < 2x + 30 < 360 and 0 < 3y - 15 < 360.
Another possible wording is, “The sum of x and y could be…”
@MITer94 @pckeller
I just made sure of the questions, and that is exactly how they are written. Those questions are from Barron’s sat math level 2.
The answer for #1 is “d” as you guys answered but I still didnt get why.
For #2 the answer is “c” but I dont know how to solve this kind of problems.
@gameplayer1234 I think #1 is a poor question and you’d be better off ignoring the fact that the answer is “D.”
You can generate all solutions to the equation csc (2x+30°) = cos (3y-15°) by rewriting it as sin (2x+30°) cos (3y-15°) = 1, then because sine and cosine range from -1 to 1 inclusive, we must have either sin (2x+30°) = cos (3y-15°) = 1, or sin (2x+30°) = cos (3y-15°) = -1. Let me know what else you don’t understand.
For 2, the only “intuitive” way for me was to construct an exponential regression, which would basically require a graphing calculator if you took it under testing conditions. If you don’t know what an exponential regression is, it’s basically the same concept as a linear regression (i.e. “line of best fit”) except you are finding the exponential function that best matches the data. However, as pckeller said, the data value for 1990 seemed a little low.
Also, shouldn’t it be “grown exponentially” (as opposed to “grown exponential”)?
@MITer94 I understood #1 but it is so hard that I have never seen it before while I was practicing. For #2, I have no graphing calculator so I have to look for an alternative. I found an equation y= pe^kt ,but I didnt know how to apply it on this question. Do you have an alternative solution without using a graphing calc.And yes it is exponentially not exponential.
@gameplayer1234 Such is life - you will encounter difficult problems (math or otherwise) that you have never seen before.
For this problem at least, notice how I immediately used sin instead of csc - it just makes life easier. A good rule of thumb is to always replace csc, sec, cot with their equivalent definitions 1/sin, 1/cos, 1/tan respectively, and that will usually simplify the problem a small amount. Also knowing the unit circle well helps.
That is just a general form of an exponential function in terms of some time variable t.
I don’t see an alternative solution that will work with certainty (unless you are really good at plotting points and estimating regressions!). I would highly suggest a graphing calculator on this problem.
But also, don’t get hung up just because you have trouble on a problem – especially one from an unofficial source. The fact is that if #2 were a real college board question, they would not have messed up the last data point. All of the successive points would have gone up by approximately the same percentage. So you could use that percentage increase to predict the next value. That’s how exponential functions actually behave and the college board will get it right in a way that will not require you to do an exponential regression.