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Problem 64 Hard Difficulty

A developmental mathematics instructor at a large university has determined that a student's probability of success 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 of mathematics passed in high school, and $a$ is the student's mathematics SAT score. She estimates that $p,$ the probability of passing the course (in percent), will be
$$
p=f(s, n, a)=0.05 a+6(s n)^{1 / 2}
$$
for $200 \leq a \leq 800,0 \leq s \leq 10,$ and $0 \leq n \leq 8$ . Assuming that the above model has some merit, find the following.
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 mathematics, a placement score of $4,$ and an SAT score of 300 .
c. Find and interpret $f_{n}(4,5,480)$ and $f_{a}(4,5,480)$ .

Answer

(A) 52.39387691
(B) 41.83281573
(C) $2.683281573,0.05$

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Video Transcript

Okay, So in this problem, instructor has found a way to calculate Ah students probability of passing a class based on variables of s and a where s is the students score on ah, departmental placement exam and is a number of semesters of mathematics passed in high school and a is the students mathematics s a T score. So I have that form your function working down here 0.5 times a plus six times s times end to 1/2 power. Now this problem has three parts A B and C part A wants you to find the probability that a student passes a class with we'll score of six on the placement exam four semesters of high school math and s a T score of 4 16 So they want you to find f of six for for 60. And all you do is simply plug in these about values for the variables into our function. So if we do that, we get 0.5 times a which is for 16 or 60 plus six times the square root of size and which is four. And if we were to multiply its all out, we get an answer of 52.39%. So this student in particular would have a 52.39% chance of passing a math class. Um, eso not party for TB. Ask you to find probability for another student. Another student this time with math. Score off. Uh, with the last score of four. Please don't score. Four with five years of 55 years, five semesters of high school math and a score of 300 it s a T. So all we gotta do again is play this into the formula. Pretty simple. Your points. You're a fine times 300 plus six times the square root of 20. All right, so if we were to calculate all this, this person has a 41.83 41.83 percent chance of feeling, So this person has more likely to fail the class and pass it. Now. Part C Part C asks you to find an interpret the partial prove it is with respect to end and a at the 0.45 480. So, if we were to take the partial derivative with respect and and find the value. At 45 for 80 we're gonna first need to take the partial derivative with respect to end. Let's go ahead and do that. If we took the derivative with respect to end, all were differentiating. Is, uh this terms? This is the only turn with on invariable in it, and everything else is constant. So if we were to differentiate that we get, we had to bring down the two. So we've got three times s times on to the native. 1/2 power times times s because off general. Right. So this is our part of the related with respect to when and if you plug in the 0.45 480 to arm for me long, we get two point 683683 And this is our partial derivative. Well, just to pay. No, no. They also want you to interpret what this value means. That we're gonna go ahead and do that. Type that here in this text box. Yes, text box. And it says, and this would be a great change. Read a change in the probability of passing her unit change in the lumber. Oh, it's a mess. Race a school while keeping well. Keeping reveals constant everything else. I mean, the other two variables. All right, so this is our This is our interpretation for the function, the partial derivative of the function with respect to end. Now, the other thing they want us to do is take the partial derivative with respect to A with respect to a at the point, the same point for five for 80. All right, so first, we're gonna need to take the partial derivative respect A and thankfully, we just have to differentiate this. So are partial derivative is simply yes. 0.5 So there are no variables. So our partial derivative at the 0.4 580 is just 0.5 If we were to interpret this, we're contemplative. Since we can't interpret this as well, um, this is just going to be a rate of change. Greater shame off. Read a change in a probability. Passing her unit change in the students in the students, Uh, s a T score. Well, keeping There you go. This is our interpretation for the partial derivative with just like to a and I believe that is it for this problem

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