On two examinations, you have grades of 87 and 89 . There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of A , meaning a final average of at least 90 . a. What must you get on the final to earn an A in the course? b. By taking the final, if you do poorly, you might risk the B that you have in the course based on the first two exam grades. If your final average is less than 80 , you will lose your B in the course. Describe the grades on the final that will cause this to happen.

To determine the grades needed on the final examination, we can set up equations based on the information provided. ### Part a: What must you get on the final to earn an A in the course? You have two exam grades: 87 and 89. The final exam will count as a third grade. To find out what you need on the final exam to achieve an average of at least 90, we can use the following formula for the average: \[ \text{Average} = \frac{\text{Grade 1} + \text{Grade 2} + \text{Final}}{3} \] Substituting the known values: \[ 90 = \frac{87 + 89 + x}{3} \] Where \( x \) is the grade on the final exam. Now, we can solve for \( x \): 1. Multiply both sides by 3: \[ 270 = 87 + 89 + x \] 2. Combine the grades: \[ 270 = 176 + x \] 3. Subtract 176 from both sides: \[ x = 270 - 176 \] \[ x = 94 \] **You must score at least 94 on the final exam to earn an A in the course.** ### Part b: Describe the grades on the final that will cause you to lose your B in the course. To determine the grades on the final that will cause your average to drop below 80, we can set up a similar equation: \[ \text{Average} = \frac{87 + 89 + x}{3} \] We want this average to be less than 80: \[ \frac{87 + 89 + x}{3} < 80 \] 1. Multiply both sides by 3: \[ 87 + 89 + x < 240 \] 2. Combine the grades: \[ 176 + x < 240 \] 3. Subtract 176 from both sides: \[ x < 240 - 176 \] \[ x < 64 \] **If you score less than 64 on the final exam, you will risk losing your B in the course.**