If a student has an average exam score between 70 and​ 79, he will earn a grade of C in his algebra course. Suppose he has three exam scores of 77​, 64​, and 83 and that his teacher said the final exam score has twice the weight of the other three exams. What range of scores on the final exam will result in him earning a grade of​ C?

To calculate the range of scores on the final exam that would result in a grade of C, we need to consider the weighted average of the exam scores.

Let \( x \) be the score on the final exam. The three exam scores are 77, 64, and 83, and the final exam has twice the weight of the other three exams.

The average score \( A \) can be calculated as follows:

\[
A = \frac{(77 + 64 + 83) + 2x}{3 + 2}
\]

Simplifying the numerator:

\[
A = \frac{224 + 2x}{5}
\]

We want this average score \( A \) to be between 70 and 79 (inclusive):

\[
70 \leq \frac{224 + 2x}{5} \leq 79
\]

### Solving the Inequalities

**1. Solving the left inequality:**

\[
70 \leq \frac{224 + 2x}{5}
\]

Multiply both sides by 5:

\[
350 \leq 224 + 2x
\]

Subtract 224 from both sides:

\[
126 \leq 2x
\]

Divide by 2:

\[
63 \leq x
\]

**2. Solving the right inequality:**

\[
\frac{224 + 2x}{5} \leq 79
\]

Multiply both sides by 5:

\[
224 + 2x \leq 395
\]

Subtract 224 from both sides:

\[
2x \leq 171
\]

Divide by 2:

\[
x \leq 85.5
\]

### Summary

Combining both parts, we have:

\[
63 \leq x \leq 85.5
\]

### Final Answer

The range of scores on the final exam that will result in the student earning a grade of C is:

\[
63 \leq x \leq 85.5
\]