To determine the range of scores on the final exam that will result in a grade of C (average between 70 and 79), we need to calculate the average considering that the final exam has twice the weight of each of the three previous exam scores.
Let:
- \( x \) be the final exam score.
- The three exam scores are \( 75, 65, \) and \( 83 \).
Since the final exam has twice the weight, the average can be calculated using the following formula:
\[ \text{Weighted Average} = \frac{\text{Exam1} + \text{Exam2} + \text{Exam3} + 2 \times \text{Final}}{3 + 2} \]
Substituting in the exam scores:
\[ \text{Weighted Average} = \frac{75 + 65 + 83 + 2x}{5} \]
Calculating the total of the first three exams:
\[ 75 + 65 + 83 = 223 \]
Now we can rewrite the average as:
\[ \text{Weighted Average} = \frac{223 + 2x}{5} \]
We want this average to be between 70 and 79. We can set up the inequality:
\[ 70 \leq \frac{223 + 2x}{5} \leq 79 \]
To eliminate the fraction, multiply all parts of the inequality by 5:
\[ 70 \times 5 \leq 223 + 2x \leq 79 \times 5 \]
This simplifies to:
\[ 350 \leq 223 + 2x \leq 395 \]
Now, we solve the two inequalities separately.
-
For the left side: \[ 350 - 223 \leq 2x \] \[ 127 \leq 2x \] \[ \frac{127}{2} \leq x \] \[ 63.5 \leq x \]
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For the right side: \[ 223 + 2x \leq 395 \] \[ 2x \leq 395 - 223 \] \[ 2x \leq 172 \] \[ x \leq \frac{172}{2} \] \[ x \leq 86 \]
So, combining both parts, we find:
\[ 63.5 \leq x \leq 86 \]
Thus, the range of scores on the final exam that will result in a grade of C is:
\[ 63.5 \leq x \leq 86 \]
You would enter this as:
- \(63.5\) in the first box (less than or equals)
- \(86\) in the second box (less than or equals).
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