Engineering: Fluid Flow, Heating, and Mixing

[maxbender]

Hello,

I've posed this problem to my engineer friends on facebook but none have provided a solution. This involves constructing a set of equations based around a set of real world constants/variables.

I've used a step-wise solution to get me to the answer, but I was wondering if any of you could help me learn how to solve this problem exactly using calculus. Problem is described in photo.

My answers using step-wise solution:
Problem 1: 21.35 minutes
Problem 2: 65.77 minutes
Problem 3: 23.230 kW

Many thanks to all of you! After I re-learn this I will be able to solve many similar problems.

 

[maxbender]

I got bored one day and decided to run some numbers on my hot water system (I know right?!), and I came up with a solution, but it was a step-wise solution that was approximate (but good enough). However I feel like I failed myself by not being able to come to an exact solution using calculus and the methods that I was taught in university. I want to learn how to do this problem using calculus, and I'm hoping that, for some of you, this will be easy and can share your method with me.

My answers using step-wise solution:
Problem 1: 21.35 minutes
Problem 2: 65.77 minutes
Problem 3: 23.230 kW

 

[stapel]

Please reply showing your work, and I'm sure someone will be glad to check it for you. Please be complete. Thank you!