a chemist has 200 ml of a 10% sucrose solution. she adds (x) ml of a 40% sucrose solution. the percent…

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}cdot100). 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?

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}cdot100). 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?

Answer

Answer:

The total volume of the final mixture is $(200 + x)$ mL.

To find the value of $x$ when $y = 30$:

  1. First, set up the equation based on the given function:
    • We have $y=\frac{0.1(200)+0.4x}{200 + x}\cdot100$, and $y = 30$. So, $30=\frac{0.1(200)+0.4x}{200 + x}\cdot100$.
    • Divide both sides of the equation by 100: $\frac{30}{100}=\frac{0.1(200)+0.4x}{200 + x}$, which simplifies to $0.3=\frac{20 + 0.4x}{200 + x}$.
  2. Then, cross - multiply:
    • $0.3(200 + x)=20+0.4x$.
    • Expand the left - hand side: $60+0.3x = 20+0.4x$.
  3. Next, solve for $x$:
    • Subtract $0.3x$ from both sides: $60=20 + 0.4x-0.3x$.
    • Simplify the right - hand side: $60=20 + 0.1x$.
    • Subtract 20 from both sides: $60 - 20=0.1x$.
    • So, $40 = 0.1x$.
    • Divide both sides by 0.1: $x=\frac{40}{0.1}=400$ mL.

Explanation:

Step1: Set up the equation

Set $y = 30$ in $y=\frac{0.1(200)+0.4x}{200 + x}\cdot100$ and simplify. $0.3=\frac{20 + 0.4x}{200 + x}$

Step2: Cross - multiply

Multiply both sides by $(200 + x)$ to get $0.3(200 + x)=20+0.4x$.

Step3: Expand and solve

Expand $0.3(200 + x)$ and then isolate $x$. $60+0.3x = 20+0.4x$, $40 = 0.1x$, $x = 400$