Practical application of algebra (hot water heater)

In algebra class in high school we sometimes complained, "When are we ever gonna use this?" Well, I do find myself using algebra sometimes, as I did today, so I thought I'd share.

I'm trying to figure out what size water heater I need for a rental house with six people, so that each person has enough hot water for a shower in the morning with these assumptions:

* One shower = 14 gallons of water total
* Cold water temperature = 67°F
* Hot water temperature = 120°F
* Desired shower water temperature = 104°F

So the formula would be:

67c + 120h
-------------- = 104
      c+h

From here I decide to figure how many gallons of hot water I need for each gallon of cold, so I set c=1, which gives us:

67 + 120h
------------- = 104
      1+h

Multiplying both sides by 1+h to get rid of the denominator on the left:

67 + 120h = 104 + 104h

Simplifying:

120h - 104h = 104 - 67

16h = 37

h = 37/16

h = 2.3125

You might not believe that I got that exact answer to four decimal places lying in bed this morning when I couldn't sleep.

Anyway, the hot water portion of the 14 gallons is 2.3125/3.3125 x 14 = 9.77 gallons, which I'll round up to 10.

So for six people, I need a tank that can deliver 6 x 10 = 60 gallons. According to the specs on a 40-gallon heater, it can deliver 68 gallons of hot water in the first hour, so a 40-gallon heater should be enough. If I were buying a new heater, I'd choose 50 gallons to be on the safe side, but the house already has a 40-gallon heater, so I'll consider that sufficient for now unless the tenants start complaining.

We had 3 people in my house with a 50 gal water heater, and the 3rd person usually got a cold shower. Perhaps we were using more than 14 gals for the the shower?

Still, I would not want to be #6.

Quote: MichaelBluejay

67c + 120h
-------------- = 104
      c+h

From here I decide to figure how many gallons of hot water I need for each gallon of cold, so I set c=1, which gives us:

Is this the way math is taught at school in the US ? Neither is c=1 part of the solution, nor does it help you. The very first step should be - look at the problem!
You already know that c+h = 6*14 = 84, which will immediately get you rid of the denominator:

67c + 120h = 104*84.

Since you don't need to store cold water, get rid of c by further substituting c = 84 - h:

67*(84 - h) + 120h = 104*84.

After substraction of simple numbers you find

(120-67)h = (104 - 67)*84

yielding h = 37*84 / 53 = 58.6....


No need to even think about the calculator except the final result.

I absolutely disagree with Algebra being able to help you buy a hot water heater...
I agree 100% with RC...

My issue is that the water leaving the hot water heater can't be guaranteed to be 120 degrees. Hot water thermostats work off the bottom of the water heater, where the cold water enters, and are not based upon the water exiting the heater at the top of the tank. It's been know that hot water temperatures leaving the thermostat can vary +/- 15 degrees.