The Correct Answer and Explanation is:
The correct answer is 1,000 pounds.
This problem can be solved by first determining the remaining stress capacity of the beam and then calculating the corresponding weight that would create that stress. The solution involves establishing a direct proportion between the applied weight and the resulting stress.
First, we find the available stress capacity. The beam has a maximum strength, or breaking point, of 3,000 psi. A weight has already been added, creating a stress of 1,000 psi. To find out how much more stress the beam can handle, we subtract the current stress from the maximum strength. The calculation is 3,000 psi (maximum) minus 1,000 psi (current), which equals 2,000 psi. This result means the beam can withstand an additional 2,000 psi of stress before it fails.
Next, we establish the relationship between weight and stress based on the information provided in the problem. The problem states that a 500-pound weight creates 1,000 psi of stress. This gives us a clear and consistent ratio. For every 1,000 psi of stress on this particular beam, a 500-pound weight must be applied at its center. This relationship is linear, meaning if you double the stress, you must also double the weight.
Finally, we use this established relationship to find the additional weight. We need to add enough weight to create the remaining 2,000 psi of stress capacity we calculated earlier. Since 2,000 psi is exactly double 1,000 psi, it logically follows that the weight required to produce this stress will also be double the original weight. Therefore, we calculate two times 500 pounds, which equals 1,000 pounds. This is the additional weight the beam can support at its center before reaching its breaking point.
