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University of Louisville



Problem 68 Hard Difficulty

Education A developmental mathematics instructor at a large
university has determined that a student's probability of suc-
cess in the university's pass/fail remedial algebra course is a
function of $s, n,$ and $a$ , where $s$ is the student's score on the
departmental placement exam, $n$ is the number of semesters
\begin{array}{c}{\text { of mathematics passed in high school, and } a \text { is the student's }} \\ {\text { mathematics SAT score. She estimates that } p \text { , the probability of }} \\ {\text { passing the course (in percent), will be }} \\ {p=f(s, n, a)=0.05 a+6(s n)^{1 / 2}}\end{array}
$$\begin{array}{l}{\text { for } 200 \leq a \leq 800,0 \leq s \leq 10, \text { and } 0 \leq n \leq 8 . \text { Assuming }} \\ {\text { that the above model has some merit, find the following. }}\end{array}$$
(a) If a student scores 6 on the placement exam, has taken
4 semesters of high school math, and has an SAT score of
$460,$ what is the probability of passing the course?
(b) Find $p$ for a student with 5 semesters of high school math-
ematics, a placement score of $4,$ and an SAT score of 300 .
(c) Find and interpret $f_{n}(4,5,480)$ and $f_{a}(4,5,480)$ .


$f_{a}(4,5,480)=0.05 \%$ the rate at which the probability of success is changing when the student's SAT score changes by one unit (from 480 to 481$)$ and the other quantities remain constant.


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Video Transcript

cool. Okay. Sixty. A developmental mathematics instructor at a large universe that has determined on a student's probability off success. You know, you're Estes Pass or fail, Matthew. Our general course is a function of s and an early where s is a student school auntie departmental placement exam and is a lumber up its master Sof masked man. It's past in high school and a is a student's mathematics and Santy score estimates. She asked him, Mr P, the probability off passing the course in person will be their function. And we have a range of a twenty, two hundred and hundreds as his front there to turn a frontier eight and way. Want Teo finds Feli A for students Score six hundred placement exam has taken force master of high school mass and high school and he's along grossness through. Okay, so any for no. Okay, we're just plugging numbers for A and equals four. Score six on the replacement exam, which is ass because six and s A T scores for sixty Shit. That's a if for sixty now a plaque in p equals zero point zero five. What flower for sixteen applause. That's a lie. Six times. What hundred six with one half. Now I just need to calculate that he's in our It was a dream of flying here. Yeah, close. Thanks. Twenty four a half. One for the zero ah, years What I ask you, but oh P's in person. That's right. Yeah. So he's going to be fifty two. Two point through knife, fifty two percent approximately. So relax for eight. Not be. It is similar, but different. A lumber. So this time five masters off high school mass, which is on he was five s equals four A equals three hundred. Ah, Now blocking P equals syrup on zero five. Twenty. Plus we're grew close. Six months. Fly forward tongues fine. Toothy one half. Now, if we can correct that. Ah, ze aeroplane. Fried flies three hundred a quarter Rosie hair appointment here does it for me in there. What a one point takes three percent. Not this car's nowhere probability to pass that eight three. Now. See, we want to find f sub end off this lumber and f sub bay. So we need to first find its up and And if something so f sub urn you cause differentiate this function with respect and eso, This party's become constant this part of we need to use Okay, we can write off how frequent their plans. Their five a day pull us six s water and one off into this part thereof thiss part constant. One of us is far channel for end one half on next to one House three. This is the one half. And to collect you one half we want to will have four five, seven for eighty Now, this is a This is can This is class. So we're plucking. We have three multiply forces. One half butterfly End was in Nephew five ninety one have now every Congress. That's just too right. So it caused six divided by a squared five. Um cock anus. These are spooks. Bye, squirt. Fine. Oh, which is Troop on fifth? Take Not That means we want to interpret it, right. That means one. A CE stays out of four estates. Before it was for eighty. When and increased by one. You got one more semester of mass, Angus, your person did go up by two point six. Yeah, so that's for FC on. Have sub a Yeah way. First find out someday equals it's just their appointed. There are five, right? This part is this constant way back in. That's all right. I hear this guy course they're part of their body. Doesn't even matter. So what? This means one your ass. And you was flying a five. Whenever you're a increased by along, that's your s a T score increased by one. Your probability off passing this art protest increased by three point zero five. Yeah. Yes, that's it for us.

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