Question

on two examinations, you have grades of 87 and 88. 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.

Explanation:

Step1: Set up the average formula for part a

Let \(x\) be the grade on the final exam. The average of the three grades \(\frac{87 + 88+x}{3}\geq90\).

Step2: Solve the inequality for part a

Multiply both sides of the inequality \(\frac{87 + 88+x}{3}\geq90\) by \(3\): \(87+88 + x\geq270\).
Simplify the left - hand side: \(175+x\geq270\).
Subtract \(175\) from both sides: \(x\geq270 - 175\).
So \(x\geq95\).

Step3: Set up the average formula for part b

We want \(\frac{87 + 88+x}{3}<80\).

Step4: Solve the inequality for part b

Multiply both sides of the inequality \(\frac{87 + 88+x}{3}<80\) by \(3\): \(87+88 + x<240\).
Simplify the left - hand side: \(175+x<240\).
Subtract \(175\) from both sides: \(x<240 - 175\).
So \(x < 65\).

Answer:

a. You must get at least a \(95\).
b. If the grade on the final \(x<65\), you will lose your B in the course.