Question

a chemist has 200 ml of a 10% sucrose solution. she adds x ml of a 40% sucrose solution. the percent concentration, y, of the final mixture is given by the rational function: y = \frac{0.1(200) + 0.4x}{200 + x} \cdot 100 the chemist needs the concentration of the final mixture to be 30%. how many milliliters of the 40% solution should she add to the 10% solution to get this concentration? \boxed{} ml what is the total volume of the final mixture? \boxed{} ml what is the mass of sucrose in the final mixture? \boxed{} g

Explanation:

Step1: Set up the equation

We know \( y = 30 \), so substitute into the formula \( y=\frac{0.1(200)+0.4x}{200 + x}\cdot100 \).
We get \( 30=\frac{0.1\times200 + 0.4x}{200 + x}\times100 \).
Simplify the numerator: \( 0.1\times200=20 \), so the equation becomes \( 30=\frac{20 + 0.4x}{200 + x}\times100 \).

Step2: Solve for \( x \)

First, divide both sides by 100: \( \frac{30}{100}=\frac{20 + 0.4x}{200 + x} \)
Simplify \( \frac{30}{100}=0.3 \), so \( 0.3=\frac{20 + 0.4x}{200 + x} \)
Multiply both sides by \( 200 + x \): \( 0.3(200 + x)=20 + 0.4x \)
Expand the left side: \( 60+0.3x = 20 + 0.4x \)
Subtract \( 0.3x \) from both sides: \( 60=20 + 0.1x \)
Subtract 20 from both sides: \( 40 = 0.1x \)
Divide both sides by 0.1: \( x = 400 \)

Step3: Find total volume

Total volume is \( 200 + x \), substitute \( x = 400 \), we get \( 200+400 = 600 \) mL.

Step4: Find mass of sucrose

Using the concentration formula, mass = concentration × volume.
Concentration is 30% = 0.3, volume is 600 mL.
So mass \( = 0.3\times600=180 \) g.

Answer:

Number of milliliters of 40% solution: 400 mL
Total volume of final mixture: 600 mL
Mass of sucrose in final mixture: 180 g