Question

se the information provided to answer Part A and Part B.

A concrete mixing tub contains 34 liters of water and 22 liters of sand. Starting at the same time, water is added to the tub at a rate of 5 liters per minute, while sand is also added to the tub at a rate of 7 liters per minute.

Part A
In how many minutes will the tub contain equal amounts of water and sand?

2 minutes
6 minutes
12 minutes
24 minutes

Part B
When the amounts of water and sand in the tub are equal, how many liters of each will be in there?

64 liters
76 liters
154 liters
170 liters

Answer 1

Part A:

Let \( W(t) \) be the amount of water in the tub at time \( t \) minutes, and \( S(t) \) be the amount of sand in the tub at time \( t \) minutes.

Initially, we have:
- Water: \( 34 \) liters
- Sand: \( 22 \) liters

Water is added at a rate of \( 5 \) liters per minute, and sand is added at a rate of \( 7 \) liters per minute. Therefore, we can express \( W(t) \) and \( S(t) \) as:

\[
W(t) = 34 + 5t
\]

\[
S(t) = 22 + 7t
\]

We want to find \( t \) when \( W(t) = S(t) \):

\[
34 + 5t = 22 + 7t
\]

Rearranging gives:

\[
34 - 22 = 7t - 5t
\]

\[
12 = 2t
\]

\[
t = 6
\]

Thus, in **6 minutes**, the tub will contain equal amounts of water and sand.

**Answer to Part A:** 6 minutes.

Part B:

Now, we need to find the amount of water and sand in the tub when they are equal. We substitute \( t = 6 \) into either \( W(t) \) or \( S(t) \):

Using the water equation:

\[
W(6) = 34 + 5(6) = 34 + 30 = 64 \text{ liters}
\]

Using the sand equation:

\[
S(6) = 22 + 7(6) = 22 + 42 = 64 \text{ liters}
\]

When the amounts are equal, there will be **64 liters** of each.

**Answer to Part B:** 64 liters.