A truck radiator holds 5 gal and is filled with water. A gallon of water is removed from the radiator and replaced with a gallon of antifreeze; then a gallon of the mixture is removed from the radiator and again replaced by a gallon of antifreeze. This process is repeated indefinitely. How much water remains in the tank after this process is repeated 3 times? 5 times? n times?
5 * .8^3
5 * .8^5
5 * .8^n
term -- water -- antifreeze
1 ........ 5 ....... 0
2 ......... (.8)5 ....... 1 ----- water is 5(.8)
3 start with 5(.8) remove .2 of that leaving 5(.8)^2 = 3.2 gallons of water
4 starts with 5(.8)^2 , remove .2 of that leaving 5(.8)^3
5 ..................... leaving 5(.8)^4
...
n ..................... leaving 5(.8)^(n-1)
go with R_scott's answer
I counted the initial stage as the first process
To solve this problem, we need to understand the process and keep track of the quantities at each step.
Initially, the truck radiator holds 5 gallons of water. In each step, we remove 1 gallon and replace it with 1 gallon of antifreeze. Let's see how the quantities change after each step.
Step 1:
- Remove 1 gallon of water: 5 gallons - 1 gallon = 4 gallons
- Add 1 gallon of antifreeze: 4 gallons + 1 gallon = 5 gallons (now it's a mixture)
Step 2:
- Remove 1 gallon of the mixture: 5 gallons - 1 gallon = 4 gallons (the removed gallon contains both water and antifreeze)
- Add 1 gallon of antifreeze: 4 gallons + 1 gallon = 5 gallons (still a mixture)
Step 3:
- Remove 1 gallon of the mixture: 5 gallons - 1 gallon = 4 gallons
- Add 1 gallon of antifreeze: 4 gallons + 1 gallon = 5 gallons
So, after the process is repeated 3 times, we end up with 5 gallons of the mixture, but the exact composition of water and antifreeze is not known. However, we can see that in each step, the total volume of the mixture remains the same at 5 gallons.
Now, let's investigate what happens after the process is repeated 5 times:
Step 4:
- Remove 1 gallon of the mixture: 5 gallons - 1 gallon = 4 gallons
- Add 1 gallon of antifreeze: 4 gallons + 1 gallon = 5 gallons
Step 5:
- Remove 1 gallon of the mixture: 5 gallons - 1 gallon = 4 gallons
- Add 1 gallon of antifreeze: 4 gallons + 1 gallon = 5 gallons
After 5 repetitions, we still have 5 gallons of the mixture. Therefore, after 5 times, the total volume of the mixture remains the same.
In general, for any positive integer n, if the process is repeated n times:
- The total volume of the mixture will always be 5 gallons.
- However, the exact composition of water and antifreeze cannot be determined without additional information.
So, after this process is repeated n times, 5 gallons of the mixture will remain in the radiator.