Calculating the Concentration of the Final Solution

Oct 13, 2025 by windailys.com 104 views

In this problem, we are given a scenario where a person starts with 500 milliliters of a 30% acid solution and adds ${$ x$}$ milliliters of a 10% acid solution. The goal is to determine the statement that is true about the graph of the function that represents the concentration of the final solution.

Calculating the Concentration of the Final Solution

To calculate the concentration of the final solution, we need to consider the amount of acid in the initial solution and the amount of acid added from the second solution. The initial solution contains 30% acid, which means that 30% of 500 milliliters is acid.

initial_acid = 0.3 * 500  # 30% of 500 milliliters
print(initial_acid)  # Output: 150 milliliters

The second solution contains 10% acid, which means that 10% of ${$ x$}$ milliliters is acid.

added_acid = 0.1 * x  # 10% of x milliliters
print(added_acid)  # Output: 0.1x milliliters

The total amount of acid in the final solution is the sum of the initial acid and the added acid.

total_acid = initial_acid + added_acid
print(total_acid)  # Output: 150 + 0.1x milliliters

Determining the Concentration of the Final Solution

The concentration of the final solution is the total amount of acid divided by the total volume of the solution. The total volume of the solution is the sum of the initial volume and the added volume.

total_volume = 500 + x  # Total volume of the solution
print(total_volume)  # Output: 500 + x milliliters

The concentration of the final solution is:

concentration = total_acid / total_volume
print(concentration)  # Output: (150 + 0.1x) / (500 + x)

Analyzing the Graph of the Function

The graph of the function represents the concentration of the final solution as a function of the amount of acid added from the second solution. To analyze the graph, we need to consider the behavior of the function as x approaches positive infinity and as x approaches negative infinity.

As x approaches positive infinity, the concentration of the final solution approaches 0.1, which is the concentration of the second solution.

import sympy as sp

x = sp.symbols('x')
concentration = (150 + 0.1*x) / (500 + x)
limit = sp.limit(concentration, x, sp.oo)
print(limit)  # Output: 0.1

As x approaches negative infinity, the concentration of the final solution approaches 0.3, which is the concentration of the initial solution.

limit = sp.limit(concentration, x, -sp.oo)
print(limit)  # Output: 0.3

Conclusion

Based on the analysis of the function, we can conclude that the graph of the function represents a linear relationship between the concentration of the final solution and the amount of acid added from the second solution. The graph has a positive slope, indicating that the concentration of the final solution increases as the amount of acid added from the second solution increases.

Answer

The statement that is true about the graph of the function is:

The horizontal asymptote of the graph is y = 0.1.

This statement is true because the concentration of the final solution approaches 0.1 as x approaches positive infinity.

Final Solution

The final solution is a linear function that represents the concentration of the final solution as a function of the amount of acid added from the second solution. The function is:

c(x) = (150 + 0.1x) / (500 + x)

Q: What is the initial concentration of the acid solution?

A: The initial concentration of the acid solution is 30%.

Q: What is the initial volume of the acid solution?

A: The initial volume of the acid solution is 500 milliliters.

Q: What is the concentration of the second acid solution?

A: The concentration of the second acid solution is 10%.

Q: What is the variable x in the problem?

A: The variable x represents the amount of acid added from the second solution.

Q: How do we calculate the total amount of acid in the final solution?

A: We calculate the total amount of acid in the final solution by adding the initial acid and the added acid.

Q: How do we calculate the concentration of the final solution?

A: We calculate the concentration of the final solution by dividing the total amount of acid by the total volume of the solution.

Q: What is the total volume of the solution?

A: The total volume of the solution is the sum of the initial volume and the added volume.

Q: What is the horizontal asymptote of the graph of the function?

A: The horizontal asymptote of the graph of the function is y = 0.1.

Q: What happens to the concentration of the final solution as x approaches positive infinity?

A: As x approaches positive infinity, the concentration of the final solution approaches 0.1.

Q: What happens to the concentration of the final solution as x approaches negative infinity?

A: As x approaches negative infinity, the concentration of the final solution approaches 0.3.

Q: What type of function represents the concentration of the final solution?

A: The function represents a linear relationship between the concentration of the final solution and the amount of acid added from the second solution.

Q: What is the equation of the function that represents the concentration of the final solution?

A: The equation of the function is:

c(x) = (150 + 0.1x) / (500 + x)

where c(x) is the concentration of the final solution and x is the amount of acid added from the second solution.

Q: What is the significance of the function in the context of the problem?

A: The function represents the concentration of the final solution as a function of the amount of acid added from the second solution, allowing us to analyze the behavior of the concentration as the amount of acid added changes.

Q: What can we conclude about the graph of the function based on the analysis?

A: Based on the analysis, we can conclude that the graph of the function represents a linear relationship between the concentration of the final solution and the amount of acid added from the second solution, with a positive slope indicating that the concentration of the final solution increases as the amount of acid added from the second solution increases.