Question

on two examinations, you have grades of 83 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 a. you must get at least a \boxed{}.

Explanation:

Step1: Define variable for final grade

Let $x$ = final exam grade.

Step2: Set up average inequality for A

Average of 3 grades ≥ 90:
$$\frac{83 + 94 + x}{3} \geq 90$$

Step3: Solve inequality for part a

Multiply both sides by 3:
$83 + 94 + x \geq 270$
Simplify left side:
$177 + x \geq 270$
Subtract 177:
$x \geq 270 - 177$
$x \geq 93$

Step4: Set up inequality for part b

Average < 80:
$$\frac{83 + 94 + x}{3} < 80$$

Step5: Solve inequality for part b

Multiply both sides by 3:
$83 + 94 + x < 240$
Simplify left side:
$177 + x < 240$
Subtract 177:
$x < 240 - 177$
$x < 63$

Answer:

a. 93
b. You must get a grade less than 63 on the final.