To solve this problem, we first need to determine the average score required for the student to earn a grade of C, which is between 70 and 79.
Let's denote the final exam score as \( x \). Since the final exam has twice the weight of the other three exams, we'll treat the final exam score as being counted twice.
Step 1: Calculating the overall average
The average score can be calculated using the formula:
\[ \text{Average} = \frac{\text{Total of scores}}{\text{Number of scores}} \]
In this case, the student has three exam scores and one final exam score counted twice. So, the total number of scores is:
\[ 3 + 2 = 5 \]
The total of the current exam scores is:
\[ 77 + 62 + 82 = 221 \]
Including the final exam score \( x \), the total will be:
\[ 221 + x + x = 221 + 2x \]
Thus, the average score will be:
\[ \text{Average} = \frac{221 + 2x}{5} \]
Step 2: Setting up the inequality
We want the average to be between 70 and 79:
\[ 70 \leq \frac{221 + 2x}{5} \leq 79 \]
Now, we can solve these inequalities.
Step 3: Solving the inequalities
- Lower Bound:
\[ 70 \leq \frac{221 + 2x}{5} \]
Multiply both sides by 5:
\[ 350 \leq 221 + 2x \]
Subtract 221 from both sides:
\[ 129 \leq 2x \]
Divide by 2:
\[ \frac{129}{2} \leq x \]
\[ x \geq 64.5 \]
- Upper Bound:
Now, for the upper bound:
\[ \frac{221 + 2x}{5} \leq 79 \]
Multiply both sides by 5:
\[ 221 + 2x \leq 395 \]
Subtract 221 from both sides:
\[ 2x \leq 174 \]
Divide by 2:
\[ x \leq 87 \]
Step 4: Conclusion
Combining both results, we find the range of scores for the final exam \( x \):
\[ 64.5 \leq x \leq 87 \]
Thus, the answer can be written as:
\[ \text{The range of scores on the final exam that will result in him earning a grade of C is } 64.5 \leq x \leq 87. \]
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