@ai solve this

A scientist needs 10 liters of a 20% acid solution for an experiment, but she has only a 5% solution and a 40% solution. About how many liters of the 5% and the 40% solutions should the scientist mix to get the solution she needs?

Choose the equation to match the situation.
A. (0.20)(10) = 0.05x + 0.40x
B. (0.20)(10) = 0.05x + 0.40(10 − x)
C. (0.20)(10) = 0.05(10) + 0.40(10 − x)
D. (0.20)(10) = 0.05(10 − x) + 0.40(10 − x)
Part B

Round the answers to the nearest tenth.

She should mix
liters of 5% solution with
liters of 40% solution.

1 answer

To determine how many liters of the 5% and 40% solutions the scientist should mix to obtain 10 liters of a 20% acid solution, we can formulate an equation based on the concentration of acid in each solution.

Let's denote:

  • \( x \) = liters of the 5% solution
  • \( 10 - x \) = liters of the 40% solution (since the total volume needs to be 10 liters)

We know:

  • The desired concentration of acid in the final solution is 20%, which means there will be \( 0.20 \times 10 = 2 \) liters of acid in the final solution.

From the two solutions, we can express the total acid in the mixture:

  • The 5% solution contributes \( 0.05x \) liters of acid.
  • The 40% solution contributes \( 0.40(10 - x) \) liters of acid.

Thus, we can write the equation: \[ 0.20 \times 10 = 0.05x + 0.40(10 - x) \]

This corresponds to option B: \[ (0.20)(10) = 0.05x + 0.40(10 - x) \]

Part B: Solving the Equation

Now let's solve for \( x \):

  1. Substitute \( 2 \) into the equation: \[ 2 = 0.05x + 0.40(10 - x) \]

  2. Distribute: \[ 2 = 0.05x + 4 - 0.40x \]

  3. Combine like terms: \[ 2 = 4 - 0.35x \]

  4. Rearrange to solve for \( x \): \[ 0.35x = 4 - 2 \] \[ 0.35x = 2 \] \[ x = \frac{2}{0.35} \] \[ x \approx 5.7143 \ (\text{liters of 5% solution}) \]

  5. Now find the amount of the 40% solution: \[ 10 - x = 10 - 5.7143 \approx 4.2857 \ (\text{liters of 40% solution}) \]

Final Answer (Rounded):

  • Liters of 5% solution: \( \approx 5.7 \)
  • Liters of 40% solution: \( \approx 4.3 \)

The scientist should mix 5.7 liters of 5% solution with 4.3 liters of 40% solution.