Question

cpm8 advanced algebra: concepts and connections - block (27.0831030)
modeling with rational functions
interpreting an intercept of a rational function that models a real - world situation
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\times50 + 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 amount of protein the feed has
if no hay is added
the weight of hay, in pounds in the
final mixture
the amount of protein the feed has
if no oats are added
the weight of oats, in pounds, in the
final mixture

Explanation:

Step1: Recall y-intercept definition

The y - intercept of a function \(y = f(x)\) occurs when \(x = 0\). So we substitute \(x = 0\) into the function \(y=\frac{0.06\times50 + 0.12x}{50 + x}\).

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

When \(x = 0\), the function becomes \(y=\frac{0.06\times50+0.12\times0}{50 + 0}=\frac{0.06\times50}{50}\).

Step3: Interpret the substitution

In the context of the problem, \(x\) represents the number of pounds of oats added. When \(x = 0\), it means no oats are added. The numerator \(0.06\times50+0.12x\) represents the total amount of protein (since \(0.06\times50\) is the protein from hay and \(0.12x\) is the protein from oats), and the denominator \(50 + x\) is the total weight of the feed mixture. When \(x = 0\), we are calculating the percent of protein in the feed mixture when only hay (and no oats) is present.

Answer:

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