Question

A completely filled 20 gallon tank originally contains 10 pounds of salt dissolved in water. pure water enters the tank at the rate of 5 gallons/minute, and the well-stirred mixture leaves the tank at the same rate. find the amount of salt in the tank at any time t .

Answer 1

To find the amount of salt in the tank at any time t, we can use the equation S(t) = 10 - 10(W(t)/20), where W(t) represents the amount of water in the tank at time t. Substituting the given values, we can calculate the amount of salt in the tank.

Step-by-step explanation:

To find the amount of salt in the tank at any time t, we need to consider the rate at which pure water enters and leaves the tank. Since water enters the tank at a rate of 5 gallons/minute, the amount of water in the tank at any time t can be represented by the equation: W(t) = 5t, where t represents time in minutes. The concentration of salt in the tank remains constant, so the amount of salt in the tank at any time t can be represented by the equation: S(t) = 10 - 10(W(t)/20), where S(t) represents the amount of salt in pounds and W(t) represents the amount of water in gallons.

Let's solve for S(t) when t = 10 minutes:
S(10) = 10 - 10(5/20) = 10 - 10(1/4) = 10 - 10/4 = 10 - 2.5 = 7.5 pounds of salt.

Answer 2

suppose you wanted the tank to hold 100 gallons of water and it was 50 gallons per cubic feet. How many cubic feet would you need? Obviously, 2 cubic feet. How did you get that?? Well, you divided 100 by 50 to get 2.

O.K., now, let's try YOUR problem. 225000 gallons and 7.5 gallons per cubic feet. How many cubic feet do you need? Must be 225,000/7.5 = 30,000 cubic feet

O.K., now we're half way there. we (should) know that the volume of a prism is

l x w x h. We now know the volume must also be 30,000

so l x w x h = 30000
100 x 20 x h = 30000

you should be able to take it from there.

Problem #2

We approach it the exact same way.

It's too hard as stated. Let's make up an easier problem.

suppose 1 gallon took up 5 cubic feet. If you had 10 cubic feet, how many gallons would you have?
well, clearly, you would have 2 gallons. How did you get it?

Now, how many cubic feet in the cylinder? Well you (should) know that the volume of a cylinder is:

Pi r^2h = 3.14 x 25^2 h = 1962.5 h

Now we have a bit of a problem. You didn't tell me how high the tank is. Let us say it is 10 feet.

So, the tank had 19,625 cubic feet.

Go back to the start of this problem and you will know what to do next.