Question

a horse owner has 50 lbs of hay that is 6% protein by weight. he adds x lbs of oats that is 12% protein by weight. the function $y = \frac{0.06(50) + 0.12x}{50 + x}$ models the percent of protein, y, in the final mixture of feed. what does the y-intercept represent in this situation? the weight of oats, in pounds, in the final mixture; the amount of protein the feed has if no oats are added; the amount of protein the feed has if no hay is added; the weight of hay, in pounds in the final mixture

Explanation:

Step1: Recall y-intercept definition

The y-intercept of a function \( y = f(x) \) is the value of \( y \) when \( x = 0 \).

Step2: Substitute \( x = 0 \) into the function

Given the function \( y=\frac{0.06(50)+0.12x}{50 + x} \), substitute \( x = 0 \):
\( y=\frac{0.06(50)+0.12(0)}{50+0}=\frac{0.06(50)}{50}=0.06 \) (or 6%).

Step3: Interpret the result

When \( x = 0 \), it means no oats (\( x \) is the weight of oats) are added. The value of \( y \) (percent protein) at \( x = 0 \) represents the percent protein in the feed when no oats are added (only hay is present). So the y-intercept represents the amount of protein the feed has if no oats are added.

Answer:

The amount of protein the feed has if no oats are added