System Of Linear Equations

Step 1: The rates of the pipes are as follows: Pipe A (inlet): of the cistern per hour. Pipe B (inlet): of the cistern per hour. Pipe C (outlet): of the cistern per hour.
Step 2: Between 5 a.m. and 6 a.m. (Only Pipe A is open): In 1 hour, Pipe A fills:
Step 3: Between 6 a.m. and 7 a.m. (Pipes A and B are open): Combined rate of Pipes A and B: In 1 hour, Pipes A and B together fill:
Step 4: At 7 a.m., Pipe C is also opened (Pipes A, B, and C are all open): Combined rate of Pipes A, B, and C: This means the cistern is being emptied at a rate of of the cistern per hour.
Step 5: Total filled by 7 a.m.: From 5 a.m. to 6 a.m. (1 hour by Pipe A): From 6 a.m. to 7 a.m. (1 hour by Pipes A and B): Total filled by 7 a.m.:
Step 6: Time to empty the cistern after 7 a.m.: At 7 a.m., the cistern is full, and it is being emptied at a rate of per hour. Time required to empty of the cistern:
Step 7: Final time: 2.2 hours after 7 a.m. is:
Final Answer: The cistern will be empty at .

Step 1: Null hypothesis : The advertising campaign was not successful, i.e., .
Step 2: Alternative hypothesis : The advertising campaign was successful, i.e., .
Step 3: Compute the -statistic:
Step 4: Compare -statistic with : Since , we reject .
Step 5: Conclusion: The advertising campaign was successful.

Step 1: Simplify the integral: Let . Substitute , so and the limits become (at ) and (at ).
Step 2: Rewrite the integral:
Step 3: Use partial fraction decomposition: Solve for and : Thus:
Step 4: Substitute back and integrate: Simplify:

Step 1: Equate the given matrices element by element:
Step 2: Solve for and from and : From , we get . Substitute into : Then .
Step 3: Solve for from :
Step 4: Solve for from : Thus, .

Step 1: In a Linear Programming Problem (LPP), if the objective function has the same value at two corner points of the feasible region, then the maximum value of will occur at all points on the line segment joining these two corner points.
Step 2: Since the line segment contains infinitely many points, the number of points at which attains its maximum value is infinite.

For , the region includes , , and , but excludes the constraints forming the region . Based on the graph, the constraints are derived as: - , - .
Final Answer: The constraints for are:

Step 1: Given one non-trivial constraint is . From the graph:
- The line passing through and corresponds to .
- The line passing through and corresponds to .
- The line passing through and corresponds to .
Final Answer: The other two non-trivial constraints are: