Tutor profile: Kristin C.
Subject: SAT II Mathematics Level 2
This question demonstrate knowledge in exponential models: Amira is an ecologist who studies the change in a particular specie's population of the Arctic Ocean over time. She found that the population drop 4.6% of its size every 2 months. The population of this specie can be modeled by function, F, which is based on the amount of time, t in months. When the study began, Amira observed that there were 90000 of this specie in the Arctic Ocean. Write a function that models the population of this specie t months since the beginning of Amira's observation.
To get my essence of what this function is to do, it is always valuable to see, to create a table for some interesting inputs for the function and seeing how the function should behave. So first of all, if t is in months, and F of t is the number of this specie in the ocean. So when t is equal to zero, there are 90000 of this specie in the ocean. Another intriguing one is we known the population decreases 4.6% every 2 months. So let's think about when t is 2, 2 months, and then the population should have gone down by 4.6%. So going down 4.6% is the same thing as retaining 95.4%. So after 2 months, you can either say the specie's population shrinks 4.6%, or you could say it is 95.4% of the population at the beginning. So after 2 months, the population should be 90000 times 0.954. Now if we go another 2 months, we are going to be 90000 times 0.954 squared. And after 6 months, we are going to be 90000 times 0.954 to the third power. And I think you might see what is going on here, we have an exponential function, between every 2 months, we are multiplying by his common ratio of 0.954. And so we could write our function F of t, our initial value is 90000, F(t)= 90000 time 0.954 to the power of however many of these 2 months period we have gone so far. On the exponential part, we divide t by 2. And so notice, when t equals zero, all of 0.954^(t/2) becomes one, and we have F(t)=90000. When t is equal to 2, this exponent is 1 (2/2=1), and we are multiplying 0.954 once; when t is 4, the exponent is going to be two, and we are going to multiply 0.954 twice by the same reasoning. And this function F(t)= 9000- * 0.954^(t/2) enable us to model the population of this observed specie.
A wheel spin, a function W(t) models the height of the topf of the wheel bar when a person has pushed the treadle for x seconds. Function: W(t)= 80-10sin(5t) What does the solution set to y= 80 - 10sin(pi/2) represent?
To understand this question, we can actually evaluate what sine of pi over two is. So sine of pi over two radians or sign of 90 degrees is going to be equal to 1 according to the unit circle. And so, that is the maximum value that this value over here can take on. Now we are going to subtract 10 times that, and that is actually the minimum value that you can take on (you cannot get any lower than -10sin(pi/2)). And so this is going to be the lowest height for the top of the wheel bar.
A result reported in the study "Outcomes at School Age After Postnatal Dexamethasone Therapy for Lung Disease of Prematurity" was that "The frequency of clinically significant disabilities was higher among children in the dexamethasone group than among controls, that 28 of 72 (39%) vs 16 of 74 (22%), p value is 0,04)."
To answer this question, we would first want to clarify the meaning of p value, that is given that our null hypothesis is true, the probability of obtaining a test statistic that is at least as extreme as the observed test statistics. Under this question's context, we know that if in fact having a significant disability at school age is independent of the treatment with dexamethasone, then the probability of observing such a different in proportions of children with disabilities between the two groups is about 4%.