Question

on two examinations, you have grades of 81 and 94. 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: Recall average formula

The average of three numbers \(x_1,x_2,x_3\) is \(\bar{x}=\frac{x_1 + x_2+x_3}{3}\). Let \(x_1 = 81\), \(x_2=94\) and \(x_3\) be the final - exam grade.

Step2: Solve for \(x_3\) to get an A

We want \(\frac{81 + 94+x_3}{3}\geq90\). First, simplify the numerator: \(81 + 94=175\). So the inequality becomes \(\frac{175 + x_3}{3}\geq90\). Multiply both sides by 3: \(175+x_3\geq270\). Then subtract 175 from both sides: \(x_3\geq270 - 175=95\).

Step3: Solve for \(x_3\) to lose a B

We want \(\frac{81 + 94+x_3}{3}<80\). Simplify the numerator: \(81 + 94 = 175\). So the inequality is \(\frac{175+x_3}{3}<80\). Multiply both sides by 3: \(175+x_3<240\). Subtract 175 from both sides: \(x_3<240 - 175 = 65\).

Answer:

a. 95
b. Grades less than 65 on the final will cause you to lose the B in the course.