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 finding the range of angles within 3 degrees of 45 degrees. The absolute value equation for a value \( x \) within \( d \) of \( a \) is \( |x - a| = d \). Here, \( a = 45 \) (the target angle) and \( d = 3 \) (the tolerance), so the equation should be \( |x - 45| = 3 \).

Step2: Solve the absolute value equation for minimum angle

To find the minimum angle, we solve \( x - 45 = - 3 \) (since we want the smaller value, we take the negative case of the absolute value).
\( x = 45 - 3 = 42 \).

Answer:

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