To find the range of scores on the final exam that will result in the student earning a grade of C (an average score between 70 and 79), we will first outline the components needed to calculate the average. The final exam has twice the weight of the other exams.
Let's define:
- \( x \) = final exam score
- The three exam scores are 77, 61, and 81.
The total contribution of the three exams is \( 77 + 61 + 81 = 219 \).
Since the final exam has twice the weight, it can be considered as contributing a score of \( 2x \).
Now we can set up the average formula:
\[ \text{Average} = \frac{(77 + 61 + 81) + 2x}{3 + 2} = \frac{219 + 2x}{5} \]
To earn a grade of C, the average must be between 70 and 79:
\[ 70 \leq \frac{219 + 2x}{5} \leq 79 \]
Step 1: Solve the lower bound (≥ 70):
\[ 70 \leq \frac{219 + 2x}{5} \]
Multiply both sides by 5:
\[ 350 \leq 219 + 2x \]
Subtract 219 from both sides:
\[ 131 \leq 2x \]
Divide by 2:
\[ 65.5 \leq x \]
Step 2: Solve the upper bound (≤ 79):
\[ \frac{219 + 2x}{5} \leq 79 \]
Multiply both sides by 5:
\[ 219 + 2x \leq 395 \]
Subtract 219 from both sides:
\[ 2x \leq 176 \]
Divide by 2:
\[ x \leq 88 \]
Conclusion:
Combining both results, we find that:
\[ 65.5 \leq x \leq 88 \]
Thus, the range of scores on the final exam that will result in him earning a grade of C is:
\[ \text{Range: } 65.5 \leq x \leq 88 \]