Question

Container A has 420 gallons of water after leaking for 10 minutes and 300 gallons of water after leaking for 25 minutes.

How many gallons per minute leak out of container A?

Answer 1

What is the starting amount of water for container A?
In other words, how much water did container A have before it started leaking water?

Answer 2

To find the rate at which water is leaking from container A, we can set up the problem using the information given about the water level at different times.

Let \( x \) be the leak rate in gallons per minute.

From the information provided:

- After 10 minutes, there are 420 gallons of water left.
- After 25 minutes, there are 300 gallons of water left.

We can express the amount of water in the container after a certain amount of time as follows:

1. After 10 minutes:
\[
\text{Amount of water} = \text{Initial amount} - \text{Leak rate} \times \text{Time}
\]
Let \( V_0 \) be the initial amount of water in the container. Then:
\[
420 = V_0 - 10x \quad \text{(1)}
\]

2. After 25 minutes:
\[
300 = V_0 - 25x \quad \text{(2)}
\]

Now we have two equations:

\[
\text{(1)} \quad 420 = V_0 - 10x
\]
\[
\text{(2)} \quad 300 = V_0 - 25x
\]

We can solve these equations to find \( x \) and \( V_0 \). First, we can solve equation (1) for \( V_0 \):
\[
V_0 = 420 + 10x \quad \text{(3)}
\]

Now, substitute equation (3) into equation (2):
\[
300 = (420 + 10x) - 25x
\]

Simplifying this gives us:
\[
300 = 420 + 10x - 25x
\]
\[
300 = 420 - 15x
\]

Now, isolate \( x \):
\[
300 - 420 = -15x
\]
\[
-120 = -15x
\]
\[
x = \frac{120}{15} = 8
\]

So, the leak rate is \( \boxed{8} \) gallons per minute.

To double-check, we can find the initial volume \( V_0 \) using \( x = 8 \):
Substituting \( x \) back into equation (3):
\[
V_0 = 420 + 10 \times 8 = 420 + 80 = 500
\]

We can verify this with both original equations:
- For equation (1):
\[
500 - 10 \times 8 = 500 - 80 = 420 \quad \text{(correct)}
\]

- For equation (2):
\[
500 - 25 \times 8 = 500 - 200 = 300 \quad \text{(correct)}
\]

Thus, the calculations are correct, and the leak rate is indeed \( \boxed{8} \) gallons per minute.

Answer 3

To find the initial amount of water in container A, we can use the leak rate we previously calculated, which is \( x = 8 \) gallons per minute, and substitute this back into one of the equations we derived.

Recall from the earlier calculations that we found:

1. From the first equation:
\[
420 = V_0 - 10x
\]

2. We found \( x = 8 \) gallons per minute.

Substituting \( x \) into the first equation:
\[
420 = V_0 - 10 \cdot 8
\]
\[
420 = V_0 - 80
\]

Now, isolate \( V_0 \):
\[
V_0 = 420 + 80
\]
\[
V_0 = 500
\]

Thus, the initial amount of water in container A before it started leaking is \( \boxed{500} \) gallons.