Question

an employee at a health food company is making a new type of trail mix. she starts with 10 lbs of hearty mix, which contains 25% dried fruit by weight. she then adds x lbs of active mix, which contains 40% dried fruit by weight. let y be the percent of dried fruit in the new type of trail mix. which function models situation? $y=\frac{0.65}{10+x} \cdot 100$ $y=\frac{0.65x}{10+x} \cdot 100$ $y=\frac{2.5 + 0.4x}{10+x} \cdot 100$ $y=\frac{0.25 + 0.4x}{10+x} \cdot 100$ complete how much active mix should she add in order to have a trail mix containing 30% dried fruit? \boxed{} lbs done

Explanation:

Step1: Recall the function model

From the problem, the function for the percentage of dried fruit \( y \) is \( y=\frac{2.5 + 0.4x}{10 + x}\cdot100 \). We want to find \( x \) when \( y = 30 \).

Step2: Substitute \( y = 30 \) into the function

Substitute \( y = 30 \) into the equation:

$$ 30=\frac{2.5 + 0.4x}{10 + x}\cdot100 $$

Step3: Simplify the equation

First, divide both sides by 100:

$$ 0.3=\frac{2.5 + 0.4x}{10 + x} $$

Then, multiply both sides by \( 10 + x \):

$$ 0.3(10 + x)=2.5 + 0.4x $$

Expand the left - hand side:

$$ 3+0.3x=2.5 + 0.4x $$

Step4: Solve for \( x \)

Subtract \( 0.3x \) from both sides:

$$ 3=2.5 + 0.1x $$

Subtract 2.5 from both sides:

$$ 0.5 = 0.1x $$

Divide both sides by 0.1:

$$ x = 5 $$

Answer:

5