Question

cphs : advanced algebra: concepts and connections - block (27.0831030)
modeling with rational functions
using a rational function to solve problems
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.
this function models the situation ( y = \frac{2.5 + 0.4x}{10 + x} cdot 100 )
how much active mix should she add in order to have a trail mix containing 30% dried fruit? (square) lbs

Explanation:

Step1: Set up the equation

We know that \( y = 30 \) (since we want 30% dried fruit), and the function is \( y=\frac{2.5 + 0.4x}{10 + x}\cdot100 \). So we set up the equation:
\( 30=\frac{2.5 + 0.4x}{10 + x}\cdot100 \)

Step2: Simplify the equation

First, divide both sides by 100:
\( \frac{30}{100}=\frac{2.5 + 0.4x}{10 + x} \)
Simplify \( \frac{30}{100} \) to \( 0.3 \):
\( 0.3=\frac{2.5 + 0.4x}{10 + x} \)

Step3: Cross - multiply

Multiply both sides by \( 10 + x \) to get rid of the denominator:
\( 0.3(10 + x)=2.5 + 0.4x \)

Step4: Distribute on the left - hand side

Using the distributive property \( a(b + c)=ab+ac \), we have:
\( 0.3\times10+0.3x = 2.5+0.4x \)
\( 3+0.3x = 2.5+0.4x \)

Step5: Solve for x

Subtract \( 0.3x \) from both sides:
\( 3=2.5 + 0.1x \)
Subtract \( 2.5 \) from both sides:
\( 3 - 2.5=0.1x \)
\( 0.5 = 0.1x \)
Divide both sides by \( 0.1 \):
\( x=\frac{0.5}{0.1}=5 \)

Answer:

5