To determine the range of scores on the final exam that would allow the student to earn a grade of C (average exam score between 70 and 79), we first need to account for the weighting of the final exam.
Let \( x \) be the score on the final exam. Since the final exam has twice the weight of the other three exams, we can model the average as follows:
- The three exam scores are: 78, 63, and 82.
- Their total sum is: \[ 78 + 63 + 82 = 223. \]
- The final exam score \( x \) counts as two scores.
Thus, the average score \( A \) can be computed using the following formula:
\[ A = \frac{78 + 63 + 82 + 2x}{3 + 2} = \frac{223 + 2x}{5}. \]
For the student to earn a grade of C, the average score must be between 70 and 79. We can set up two inequalities to find the values of \( x \):
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For the lower bound (70): \[ \frac{223 + 2x}{5} \geq 70. \] Multiply both sides by 5: \[ 223 + 2x \geq 350. \] Subtract 223 from both sides: \[ 2x \geq 127. \] Divide both sides by 2: \[ x \geq 63.5. \]
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For the upper bound (79): \[ \frac{223 + 2x}{5} \leq 79. \] Multiply both sides by 5: \[ 223 + 2x \leq 395. \] Subtract 223 from both sides: \[ 2x \leq 172. \] Divide both sides by 2: \[ x \leq 86. \]
Combining both results, we find:
\[ 63.5 \leq x \leq 86. \]
Therefore, the range of scores on the final exam that will result in the student earning a grade of C is:
63.5 ≤ x ≤ 86.