Final answer:
The probability that neither friend needs to wait more than ten minutes for the other when meeting between 6:00 and 6:30 pm is calculated by dividing the area representing acceptable arrival times by the total area of the time window. This results in a probability of 44.44%.
Step-by-step explanation:
The question at hand involves calculating the probability that neither of two friends will have to wait more than ten minutes for the other when they meet at a designated spot between 6:00 and 6:30 pm. This is a uniform distribution problem where the two friends arrive at any time during the 30-minute window with equal likelihood.
To solve this problem, we can represent the arrival times of the two friends on a coordinate grid where the x-axis represents the arrival time of the first friend and the y-axis represents the arrival time of the second friend between 6:00 and 6:30 pm. The area where neither friend waits more than ten minutes for the other will be a square taken from the diagonal of the larger square (which represents the entire 30-minute window for both). This is because if one friend arrives at a particular time, the other needs to arrive within ten minutes of that time to ensure no one waits longer than ten minutes.
If we consider a 30-minute period, the area representing acceptable arrival times for both friends forms a square on the coordinate plane where each side is 20 minutes long (the 30 minutes minus the 10-minute maximum wait). To calculate the probability, we divide the area of the smaller square (20 minutes * 20 minutes) by the area of the larger square (30 minutes * 30 minutes). Therefore, the probability is (20 * 20) / (30 * 30), which simplifies to 4/9 or approximately 0.4444. Hence, there is a 44.44% probability that neither friend will wait more than ten minutes.