Building A and building B are 500 meters apart. There is no road between them, so to drive from building [tex]$A$[/tex] to building B, it is necessary to first drive to building C and then to building B.
About how much farther is it to drive than to walk directly from building A to building B? Round to the nearest whole number.
183 meters
250 meters
366 meters
683 meters
Answer (1)
Analyze the problem and identify the given information: the direct distance between buildings A and B is 500 meters.
Recognize that the driving distance is the sum of the distances from A to C and C to B, which is greater than the direct distance due to the triangle inequality.
Consider possible scenarios for the location of building C and estimate the extra driving distance.
Select the most reasonable estimate from the given options: 366 meters.
Explanation
Problem Analysis Let d ( A , B ) be the distance between building A and building B, which is given as 500 meters. Let d ( A , C ) be the distance between building A and building C, and d ( C , B ) be the distance between building C and building B. The driving distance is d ( A , C ) + d ( C , B ) . The difference between the driving distance and the walking distance is d ( A , C ) + d ( C , B ) − d ( A , B ) . We need to find the approximate value of this difference from the given options.
Triangle Inequality Let x be the extra distance, so d ( A , C ) + d ( C , B ) = d ( A , B ) + x = 500 + x . Since A, B, and C form a triangle, by the triangle inequality, d(A, B)"> d ( A , C ) + d ( C , B ) > d ( A , B ) , which means 500"> 500 + x > 500 , so 0"> x > 0 . This condition is satisfied by all the given options.
Possible Scenarios We need to determine which of the given options is the most reasonable estimate for the extra distance. Without more information about the location of building C, we can't determine the exact extra distance. However, we can consider a few scenarios.
If C is on the line connecting A and B, then the extra distance would be 0, which is not possible since there is no road between A and B.
If the angle ACB is close to 0, then the extra distance will be large. If the angle ACB is close to 180 degrees, the extra distance will be small.
Estimation Since we don't have enough information to determine the exact value, we can only make an educated guess. The options are 183, 250, 366, and 683. If we assume that the angle ACB is not very small, then the extra distance should not be too large. Among the given options, 183, 250, and 366 seem more reasonable than 683. Without additional information, it's difficult to choose the best answer. However, since we are asked for an approximate value, we can consider that the driving distance is likely to be significantly longer than the direct distance, but not drastically so. Therefore, 366 seems like a reasonable estimate.
Final Answer Based on the analysis, the most reasonable estimate for the extra distance is 366 meters.
Examples
Imagine you are planning a road trip. You know the direct distance between two cities, but there's a mountain in the way, so you have to drive around it. Estimating how much farther you'll drive than if you could go directly helps you plan your fuel and time accordingly. This is similar to the problem, where building C represents a detour.
