Answer:
To solve this problem, we need to determine how much lemonade mix and water will be in the tank at a given time after the dispensing machine has been running for a certain amount of time.
Let's start by figuring out the rate of change of the water and lemonade mix in the tank. The rate of change of the water in the tank is 10 gallons per minute, and the rate of change of the lemonade mix in the tank is 6 pounds per minute.
So if we let t represent the number of minutes that the dispensing machine has been running, then the amount of water in the tank after t minutes will be:
200 + 10t gallons
And the amount of lemonade mix in the tank after t minutes will be:
105 + 6t pounds
Now, let's say we want to know how much lemonade we can make with the current mixture in the tank. To do this, we need to know the ratio of water to lemonade mix. We can find this by dividing the amount of water by the amount of lemonade mix:
(200 + 10t) / (105 + 6t)
This gives us the ratio of water to lemonade mix at any given time t.
Let's say we want to make 500 gallons of lemonade. We can set up an equation to solve for the amount of time it will take to make that much lemonade:
500 = (200 + 10t) / (105 + 6t)
To solve for t, we can cross-multiply:
500(105 + 6t) = 200 + 10t
52,500 + 3,000t = 200 + 10t
2,800t = 52,300
t ≈ 18.68
So it will take approximately 18.68 minutes to make 500 gallons of lemonade with the current mixture in the tank.
Explanation:
Step 1: Identify the given information and what we need to find
We are given that we have a mixing tank with 200 gallons of water and 105 pounds of powdered lemonade mix. We also know that we can add water to the tank at a rate of 10 gallons per minute and powdered lemonade mix at a rate of 6 pounds per minute. We need to determine how much lemonade we can make with the current mixture in the tank and how long it will take to make that much lemonade.
Step 2: Determine the rate of change of water and lemonade mix in the tank
Since we are adding water to the tank at a rate of 10 gallons per minute and powdered lemonade mix at a rate of 6 pounds per minute, the amount of water and lemonade mix in the tank will increase by those amounts for each minute that the dispensing machine is running.
So after t minutes, the amount of water in the tank will be:
200 + 10t
And the amount of lemonade mix in the tank will be:
105 + 6t
Step 3: Determine the ratio of water to lemonade mix
To find the ratio of water to lemonade mix in the tank at any given time t, we need to divide the amount of water by the amount of lemonade mix:
(200 + 10t) / (105 + 6t)
This gives us the ratio of water to lemonade mix at any given time t.
Step 4: Set up an equation to solve for the time it will take to make 500 gallons of lemonade
We are told that we want to make 500 gallons of lemonade. To solve for the time it will take to make 500 gallons of lemonade, we can set up an equation:
500 = (200 + 10t) / (105 + 6t)
Step 5: Solve for t
To solve for t, we can cross-multiply the equation:
500(105 + 6t) = 200 + 10t
52,500 + 3,000t = 200 + 10t
2,800t = 52,300
t ≈ 18.68
So it will take approximately 18.68 minutes to make 500 gallons of lemonade with the current mixture in the tank.