Expert: Sherman D. - 10/4/2004

Question
Kai realizes that a test grade is a function of the number of hours studied and knows from past experience that 1 hour of studying will result in a grade of 66;2 hours, in a grade of 75;and 5 hours, in a grade of 90.
a. If Kai's grade is a quadratic function,f, of the number of hours studied,x,determine an equation for f(x).
b. What is the minimum amount of time Kai must study to get a grade of at least 80?
c. According to the function in part a, what is the maximum grade that Kai can get on the exam?
The answers are:
a. f(x)=-x^2+12x+55  
b. 2hr 41 min
c. 91
I can't get any of those answers. Can you please explain.Thank you so much.

Answer
(1,66), (2,75), (5,90)

f(x) = ax^2 + bx + c

66 = a + b + c
75 = 4a + 2b + c
90 = 25a + 5b + c

a = -b - c + 66
75 = 4(-b - c + 66) + 2b + c
90 = 25(-b - c + 66) + 5b + c

75 = -4b - 4c + 264 + 2b + c
90 = -25b - 25c + 1650 + 5b + c

75 = -2b - 3c + 264
90 = -20b - 24c + 1650

2b + 3c = 189
20b + 24c = 1560

Multiply top by -8

-16b - 24c = -1512
20b + 24c = 1560

4b = 48
b = 12

now plug that in for "b"

2b + 3c = 189
2(12) + 3c = 189
24 + 3c = 189
3c = 165
c = 55

now plug those both in for "b" and "c" in a = -b - c + 66 and you get

a = -12 - 55 + 66
a = -67 + 66
a = -1

so this gave you
a = -1
b = 12
c = 55

Now plug those values in for f(x) = ax^2 + bx + c, and you get

f(x) = -x^2 + 12x + 55

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b.)
f(x) = -x^2 + 12x + 55
80 = -x^2 + 12x + 55
-x^2 + 12x - 25 = 0
-(x^2 - 12x + 25) = 0
using the quadratic formula
x = (-b ± sqrt(b^2 - 4ac))/2a
x = (-(-12) ± sqrt((-12)^2 - 4(1)(25)))/2(1)
x = (12 ± sqrt(144 - 100))/2
x = (12 ± sqrt(44))/2
x = (12 ± sqrt(4 * 11))/2
x = (12 ± 2sqrt(11))/2
x = 6 ± sqrt(11)
x = 2.68337520964... or 9.316624790355...
since they want the minimum time, take 2.68337520964 and transfer it into time
x = 2 + (.68337520964 * 60)
x = 2 + 41.002512578676...
or just say
x = 2 hours 41 minutes

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c.)
f(x) = -x^2 + 12x + 55
x = -b/2a
x = -12/(-2)
x = 6
f(x) = -(6)^2 + 12(6) + 55
f(x) = -36 + 72 + 55
f(x) = 36 + 55
f(x) = 91

and that is how they got the answers.

don't understand, just let me know.

If you are the one that asked if i could help you with other math questions, if so, you know where to find me, lol.