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. $|k-3|=45$; 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
Answer (1)
The problem is to find the equation representing the optimal launch angle range and the minimum optimal angle. The launch angle x must satisfy ∣ x − 45∣ = 3 . Solving this gives 42 l e x l e 48 . The minimum optimal angle is 42 degrees.
Explanation
Setting up the equation Let x be the launch angle. The problem states that the optimal launch angle is within 3 degrees of 45 degrees. This can be expressed as an absolute value inequality: ∣ x − 45∣ l e 3 .
Solving the inequality To find the minimum and maximum optimal angles, we solve the inequality. The inequality ∣ x − 45∣ l e 3 is equivalent to − 3 l e x − 45 l e 3 .
Finding the range of optimal angles Adding 45 to all parts of the inequality, we get − 3 + 45 l e x l e 3 + 45 , which simplifies to 42 l e x l e 48 .
Determining the minimum angle and the equation The minimum optimal angle is the lower bound of this interval, which is 42 degrees. The equation that represents the condition is ∣ x − 45∣ = 3 .
Examples
Understanding angle ranges is crucial in many real-world applications. For example, in sports like basketball, knowing the optimal launch angle for a free throw can significantly increase the chances of making the shot. Similarly, in engineering, designing a bridge requires precise angle calculations to ensure stability and weight distribution. By mastering these concepts, you can apply them to improve performance in sports or contribute to building safer and more efficient structures.
