Question

kari 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 for range

The problem is about an angle \( x \) being within 3 degrees of 45 degrees. The absolute value equation for a value \( x \) within \( d \) of a number \( a \) is \( |x - a|=d \). Here, \( a = 45 \) and \( d=3 \), so the equation is \( |x - 45| = 3 \).

Step2: Solve for minimum angle

To find the minimum angle, we solve the absolute value equation. The equation \( |x - 45| = 3 \) means \( x-45 = 3 \) or \( x - 45=- 3 \). For the minimum angle, we take the negative case: \( x-45=-3 \). Solving for \( x \), we add 45 to both sides: \( x=45 - 3=42 \).

Answer:

\( |x - 45| = 3 \); minimum angle: 42 degrees (corresponding to the third option: \( |x - 45| = 3 \); minimum angle: 42 degrees)