To find the range of final exam scores that will allow the student to earn a grade of C, we first need to average the scores of all the exams, taking into account that the final exam has twice the weight of the other three exams.
Let:
- Exam 1 score = 76
- Exam 2 score = 61
- Exam 3 score = 82
- Final exam score = \( x \)
Since the final exam counts as twice the weight, we will have a total of 5 scores to average (3 regular scores and the final exam counted twice). The contribution of the exams to the average will be calculated as follows:
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Sum of the first three exams: \[ 76 + 61 + 82 = 219 \]
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Total score including final exam: \[ \text{Total score} = 219 + x + x = 219 + 2x \]
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Total number of grades (with the weighted final): \[ 3 + 2 = 5 \]
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Average score: \[ \text{Average} = \frac{219 + 2x}{5} \]
To find the range of scores on the final exam \( x \) that will yield an average between 70 and 79, we will set up two inequalities:
Inequality for 70:
\[ \frac{219 + 2x}{5} \geq 70 \] \[ 219 + 2x \geq 350 \] \[ 2x \geq 350 - 219 \] \[ 2x \geq 131 \] \[ x \geq \frac{131}{2} = 65.5 \]
Inequality for 79:
\[ \frac{219 + 2x}{5} \leq 79 \] \[ 219 + 2x \leq 395 \] \[ 2x \leq 395 - 219 \] \[ 2x \leq 176 \] \[ x \leq \frac{176}{2} = 88 \]
Conclusion:
Thus, the range of scores on the final exam that will result in the student earning a grade of C is: \[ 65.5 \leq x \leq 88 \]
You can write this as: \[ \text{Range: } 65.5 \leq x \leq 88 \]
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