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?

A$|x-3|=45$; minimum angle: 42 degrees
B$|x-3|=45$; minimum angle: 45 degrees
C$|x-45|=3$; minimum angle: 42 degrees
D$|x-45|=3$; minimum angle: 45 degrees

Question

amel65 amel65

03-11-2026
Mathematics

Answered

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?

A$|x-3|=45$; minimum angle: 42 degrees
B$|x-3|=45$; minimum angle: 45 degrees
C$|x-45|=3$; minimum angle: 42 degrees
D$|x-45|=3$; minimum angle: 45 degrees

Verified Answer :

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Translate the condition "within 3 degrees of 45 degrees" into the absolute value equation $|x - 45| = 3$.
Solve the equation for two boundary cases: $x - 45 = 3$ and $x - 45 = -3$.
Identify the minimum optimal angle as $x = 42$.
The correct option is $|x-45|=3$; minimum angle: $42$ degrees.
Final Answer: $\boxed{|x-45|=3; \text{minimum angle: } 42 \text{ degrees}}$

Explanation

Analysis of the Problem

The problem specifies that the optimal launch angle $x$ for the water-balloon launcher is within 3 degrees of 45 degrees. In mathematical terms, the "distance" between the launch angle $x$ and the target angle of 45 degrees must be at most 3 degrees. This distance is represented by the absolute difference $|x - 45|$. The condition "within 3 degrees" translates to the inequality $|x - 45| \leq 3$. To find the minimum and maximum optimal angles, we set this absolute difference to its boundary value: $|x - 45| = 3$.

Solving the Equation for Optimal Angles

To find the range of optimal angles, we solve the equation $|x - 45| = 3$. This absolute value equation splits into two possible cases:

$x - 45 = 3$
$x - 45 = -3$

Solving the first case:
$x = 45 + 3 = 48$

Solving the second case:
$x = 45 - 3 = 42$

Thus, the optimal launch angles are between 42 and 48 degrees, inclusive. The minimum optimal angle is 42 degrees, and the maximum optimal angle is 48 degrees.

Conclusion and Final Answer

Comparing our findings with the given options, we identify the correct equation as $|x - 45| = 3$. The minimum angle derived from this equation is 42 degrees. Therefore, the correct choice is the third option: $|x-45|=3$; minimum angle: 42 degrees.

Examples

In engineering and sports, tolerances are crucial for performance. Just as in this balloon-launching example, if a machine component must be machined to a specific dimension with a certain tolerance, or if an athlete must aim within a specific range to hit a target, absolute value equations help define these acceptable ranges. For example, if a machine part is designed to be 10 cm with a tolerance of 0.05 cm, it can be modeled by the equation $|x - 10| = 0.05$, identifying that the component is acceptable if it measures between 9.95 cm and 10.05 cm. Understanding these ranges helps in quality control and precision tasks in various fields.