in real life in a GP you would need to decide in 10 minutes whether needs admission and if so why. it can be a simple urine infection but this sounds like more. the patient is old and delirious. renal pain. in GP land even if you think he needs hospital you have to then decide if this is 'medical' and goes to the AAU, if it is 'surgical' and goes to the urologist or if it is 'surgical' and goes to the general surgeons.the fact he is acutely confused (delerious), lives alone and his wife died, has a high fever, has signs on examination of possible pyelonephritis (renal pain) would make me think admission was required.whehter or not this is UTI or pyelonephritis, no one know. old and confused with a fever could really be anything. it could be UTI, it could be renal pathology, it could something else renal like a stone. it be bowel related - appendicitis, bowel obstruction. it could be gynae if it was a woman, it could even be referred pain from bones.
you need to do a proper examination first, which would guide you. bloods and imaging are rarely helpful unless you know what you are looking for and have good rational for it. in this case in GP the urine test would be helpful. if it was full of blood it would go to the urologists. if negative i think medics.
appendicitis you can diagnose on examination alone. i have never heard of imaging for appendicitis. you cant even see appendicitis on an image. the treatment is take it out before it bursts.