Question

on two examinations, you have grades of 88 and 84. 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.
a. you must get at least a

Explanation:

Part (a)

Step1: Define variables and average formula

Let \( x \) be the grade on the final exam. The average of three grades (88, 84, and \( x \)) is given by the formula for the mean: \(\text{Average} = \frac{88 + 84 + x}{3}\). We want the average to be at least 90, so we set up the inequality: \(\frac{88 + 84 + x}{3} \geq 90\).

Step2: Simplify the left - hand side

First, calculate \( 88+84 = 172 \). So the inequality becomes \(\frac{172 + x}{3} \geq 90\).

Step3: Solve for \( x \)

Multiply both sides of the inequality by 3 to get rid of the denominator: \(172 + x\geq90\times3\). Since \(90\times3 = 270\), the inequality is \(172+x\geq270\). Then subtract 172 from both sides: \(x\geq270 - 172\). Calculate \(270 - 172=98\).

Part (b)

Step1: Define the average formula for the "losing B" case

We want the average of the three grades to be less than 80. Using the same variable \( x \) for the final exam grade, the average formula gives us the inequality: \(\frac{88 + 84+x}{3}<80\).

Step2: Simplify the left - hand side

As before, \(88 + 84=172\), so the inequality is \(\frac{172 + x}{3}<80\).

Step3: Solve for \( x \)

Multiply both sides by 3: \(172+x < 80\times3\). Since \(80\times3 = 240\), we have \(172+x < 240\). Subtract 172 from both sides: \(x < 240 - 172\). Calculate \(240 - 172 = 68\). So if the grade on the final exam is less than 68, you will lose your B.

Answer:

s:
a. You must get at least a \(\boldsymbol{98}\) on the final.

b. If you get a grade less than \(\boldsymbol{68}\) on the final, you will lose your B.