The more you study for a certain exam, the better your performance on it. If you study for 10 hours, your score will be 65%. If you study for 20 hours, your score will be 95%.You can get as close as you want to a perfect score just by studying long enough. Assume your percentage score, p(n), is a function of the number of hours, n, that you study in the form:
p(n)=(an+b)/(cn+d).
If you want a score of 80%, how long do you need to study?
(This problem involves with rational function chapter)
Since the limit of p(n) is 1, a = c. So, you can scale things so that a=c=1 and you have
p(n) = (n+b)/(n+c)
Now plug in your numbers:
(10+b)/(10+c) = .65
(20+b)/(20+c) = .95
solve for b and c and you have
p(n) = (n - 107/12)/(n - 100/12)
or, eliminating the fractions,
p(n) = (12n-107)/(12n-100)
So, for a score of 80%,
(12n-107)/(12n-100) = .80
n = 11.25
p(n) = (n-b)/(n+c)
Sorry, should have been (n-b)/(n-d)
In any case, when you solve for the two unknowns, you get -107/12 and -25/3
Thanks for replying Steve
I just wanted to ask how did you get p(n) = (n - 107/12)/(n - 100/12) after solving for b and c?
Well, if we want a score of 80%, we can set up the equation p(n) = 80 and solve for n.
Given that p(n) = (an+b)/(cn+d), let's substitute the given values for n=10 and n=20:
For n = 10, p(10) = (a * 10 + b) / (c * 10 + d) = 65%
For n = 20, p(20) = (a * 20 + b) / (c * 20 + d) = 95%
Now, let's set up a system of linear equations with these two equations:
65 = (10a + b) / (10c + d)
95 = (20a + b) / (20c + d)
To solve this system, we need to eliminate either a or b. Let's eliminate a:
Multiply the first equation by 2 and subtract the second equation from it:
130 = (20a + 2b) / (10c + d)
95 = (20a + b) / (20c + d)
130 - 95 = (20a + 2b) / (10c + d) - (20a + b) / (20c + d)
35 = (20a + 2b) (20c + d) - (20a + b) (10c + d) / (10c + d) (20c + d)
By simplifying this expression, we get:
35 = 20bc + 2bd - 200c - 20d
Rearranging the terms:
20bc - 200c = 2bd - 20d - 35
Now we can factor out c and d:
20c(b - 10) = 2d(b - 10) - 35
Since we're looking for a value of n where the score is 80%, we know that b = 80. So we can substitute b = 80 into the equation:
20c(80 - 10) = 2d(80 - 10) - 35
20c(70) = 2d(70) - 35
Now we can divide both sides of the equation by 70 to simplify it:
20c = 2d - 35/70
20c = 2d - 1/2
40c = 4d - 1
40c + 1 = 4d
Now we have an equation with only d in it. Let's solve for d:
d = (40c + 1) / 4
So, the number of hours we need to study to get a score of 80% is given by the value of n when p(n) = 80. Substituting the expression for d:
n = (an + 80) / (c * n + (40c + 1) / 4)
Now, I must admit, this equation looks a bit daunting. But don't worry, I'm here to make you smile, not to solve complicated algebraic equations! Keep studying, and remember, laughter is the best study break!
To solve this problem, we need to find the values of a, b, c, and d in the given rational function p(n) = (an + b)/(cn + d) that will give us a score of 80% when we substitute the number of hours studied, n.
Given that when studying for 10 hours, the score is 65% and when studying for 20 hours, the score is 95%, we can use these two data points to form a system of equations.
First, let's solve for a and b using the data point (10, 65%):
65% = (10a + b)/(10c + d)
Multiplying both sides of the equation by (10c + d) to eliminate the denominator:
0.65(10c + d) = 10a + b
Expanding the equation:
6.5c + 0.65d = 10a + b (Equation 1)
Next, let's solve for a and b using the data point (20, 95%):
95% = (20a + b)/(20c + d)
Multiplying both sides of the equation by (20c + d) to eliminate the denominator:
0.95(20c + d) = 20a + b
Expanding the equation:
19c + 0.95d = 20a + b (Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) with two unknowns (a and b).
To solve this system, we can subtract Equation 1 from Equation 2 to eliminate a and b:
19c + 0.95d - (6.5c + 0.65d) = 20a + b - (10a + b)
Simplifying the equation:
12.5c + 0.3d = 10a
Dividing both sides of the equation by 10:
1.25c + 0.03d = a (Equation 3)
Now we have an expression for a in terms of c and d.
We can substitute this expression for a into Equation 1 to solve for b:
6.5c + 0.65d = 10(1.25c + 0.03d) + b
Expanding and simplifying the equation:
6.5c + 0.65d = 12.5c + 0.3d + b
6.5c + 0.65d - 12.5c - 0.3d = b
(-6c + 0.35d) = b (Equation 4)
Now we have expressions for both a and b in terms of c and d.
Next, we can substitute the values of a and b into the original rational function and set it equal to 80% (0.8). We will solve for n, the number of hours studied:
p(n) = (an + b)/(cn + d)
0.8 = [(1.25c + 0.03d)n + (-6c + 0.35d)]/(cn + d)
Cross-multiplying to eliminate the denominator:
0.8(cn + d) = (1.25cn + 0.03dn - 6c + 0.35d)n + (-6c + 0.35d)
Expanding and simplifying the equation:
0.8cn + 0.8d = 1.25cn^2 + 0.03dn^2 - 6cn + 0.35dn - 6cn + 0.35d
1.25cn^2 + 0.03dn^2 - 13.8cn + 0.7dn - 0.8cn - 0.35dn + 6cn - 0.35d - 0.8cn - 0.8d = 0
Combining like terms:
1.25cn^2 + 0.03dn^2 - 8.75cn + 0.3dn - 1.15d = 0 (Equation 5)
Now, we have a quadratic equation (Equation 5) that represents the relationship between the number of hours studied (n) and the desired score (p(n) = 80%).
To find the solution for n, you can solve this quadratic equation using various methods (e.g., factoring, completing the square, quadratic formula).
Once you find the values of n, you will know how long you need to study to achieve a score of 80%.