To find out what score Anna needs on her fifth quiz to achieve a mean of 85, we first need to calculate the total score she needs for all five quizzes.
The formula for the mean (average) score is:
\[ \text{Mean} = \frac{\text{Total Score}}{\text{Number of Quizzes}} \]
Given that she wants a mean of 85 after five quizzes, we can express this as:
\[ 85 = \frac{\text{Total Score}}{5} \]
To find the total score needed, we multiply both sides by 5:
\[ \text{Total Score} = 85 \times 5 = 425 \]
Next, we calculate the total of her current scores on the first four quizzes:
\[ 78 + 86 + 90 + 82 = 336 \]
Now, let’s denote her fifth quiz score as \( x \). We want the total score of all five quizzes to be 425, so we set up the equation:
\[ 336 + x = 425 \]
Now, solving for \( x \):
\[ x = 425 - 336 = 89 \]
Therefore, Anna needs to score 89 on her fifth quiz to achieve a mean quiz score of 85.
So the correct response is 89.