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Math homework help?
The enrollment of public schools is P(t)= -t^4 + 10t^3+ 3t^2-8t+1300 and the enrollment of a neighboring catholic school is C(t)= -t^4 + 8t^3 + 12t^2 + 1285. Over a ten year period, when will the public school enrollment exceed the catholic school enrollment?
How would I answer this question without graphing it?
2 Answers
- ?Lv 76 years ago
Get D(t) = P(t) - C(t) and get t such that P(t) - C(t) = 0
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D(t) = P(t)-C(t) = [ -t^4 +10t^3 +3t^2 -8t +1300 ] - [-t^4 +8t^3+12t^2+1285 ]
D(t) = P(t) - C(t) = -t^4 + t^4 + 10t^3 -8t^3 +3t^2 - 12t^2 - 8t + 1300 - 1285
D(t) = P(t) - C(t ) = 2t^3 - 9t^2 - 8t + 15
Let t = 0 :
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D(t) = P(t) - C(t) = 1300 - 1285 = 15
The public school enrollment exceeds the catholic school enrollment at the present time. <-----------------
Consider t = 1 yr:
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D(t) = ( 2 )(1)^3 - ( 9 )(1 )^2 - ( 8 ) ( 1 ) + 15
D(t) = 2 - 9 - 8 + 15 = 0
The public school and catholic school will have the same enrollment after 1 year. <------
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- 6 years ago
The moment they have the same enrollment.
-t^4 + 10t^3+ 3t^2-8t+1300 = -t^4 + 8t^3 + 12t^2 + 1285
2t^3 - 9t^2 - 8t + 15 = 0
2(1)^3 - 9(1)^2 - 8(1) + 15 = 0
(t-1) is a factor.
Divide 2t^3 - 9t^2 - 8t + 15 by (t-1) using euclid/horner and find a 2nd degree polynomial, which you can factor again. You should get :
(2t + 3)(t-1)(t-5) = 0
When t = -3/2 or t = 1 or t = 5
t = -3/2 is obviously rejected.
So at t=1 and t=5, one exceeds the other