Question

kan and samantha have determined that their water - balloon launcher works best when they launch the balloon at an angle within 3 degrees of 45 degrees. which equation can be used to determine the minimum and maximum optimal angles of launch, and what is the minimum angle that is still optimal?
|x - 3| = 45; minimum angle: 42 degrees
|x - 3| = 45; minimum angle: 45 degrees
|x - 45| = 3; minimum angle: 42 degrees
|x - 45| = 3; minimum angle: 45 degrees

Explanation:

Step1: Understand absolute - value equation concept

The general form of an absolute - value equation for the distance between a variable \(x\) and a number \(a\) is \(|x - a|=b\), where \(b\) represents the distance. Here, the ideal angle is 45 degrees and the acceptable deviation is 3 degrees. So the equation representing the angles \(x\) within 3 degrees of 45 degrees is \(|x - 45| = 3\).

Step2: Solve the absolute - value equation

The absolute - value equation \(|x - 45|=3\) can be written as two separate equations: \(x−45 = 3\) or \(x−45=-3\).
For \(x−45 = 3\), we get \(x=45 + 3=48\).
For \(x−45=-3\), we get \(x=45-3 = 42\).

Answer:

C. \(|x - 45| = 3\); minimum angle: 42 degrees