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An experienced teaching assistant can grade all exams for a calculus class 2 hours faster than a new teaching assistant. Working together, the teaching assistants can grade all the exams for the calculus class in 7 hours. How long would it take for the new teaching assistant to grade all of the exams for the calculus class alone?

User ABCDEFG
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20 votes
20 votes

Answer:

15.07 hours

Explanation:

The given relationship lets us write an equation in that uses rates in terms of tasks per hour. Let 'a' represent the number of hours it takes the assistant to grade exams alone. Then (a-2) is the number of hours it takes the experienced teacher. Working together, their total rate is ...

1/a + 1/(a-2) = 1/7 . . . . . . . tasks per hour

7(a -2) +7(a) = a(a -2) . . . . . multiply by 7a(a -2)

14a -14 = a^2 -2a . . . . . . . eliminate parentheses

a^2 -16a = -14 . . . . . . . . . subtract 14a

a^2 -16a +65 = 50 . . . . add 64 to complete the square

(a -8)^2 = 50 . . . . . . . write as a square

a -8 = ±5√2 . . . . . . take the square root

a = 8 +5√2 ≈ 15.072

It would take the teaching assistant about 15.07 hours to grade the exams alone.

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Additional comment

The quadratic also gives an extraneous solution of about 0.93 hours. This results in a negative task time for the experienced teacher. That solution makes no sense in the given scenario, so we ignored it.

User Louielyl
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