Question

1. formulate a 1500 lb feed ration for lactating dairy cows that require a crude protein of 15%. the two feedstuffs to include are corn at 8.9% and alfalfa hay at 11.9%.

Explanation:

Step1: Set up variables

Let \(x\) be the amount of corn (in lb) and \(y\) be the amount of alfalfa hay (in lb). We have two equations: \(x + y=1500\) (total weight) and \(0.089x + 0.119y=0.15\times1500\) (protein content).
From \(x + y = 1500\), we get \(x=1500 - y\).

Step2: Substitute into the protein equation

Substitute \(x = 1500 - y\) into \(0.089x+0.119y = 225\) (since \(0.15\times1500 = 225\)).
\(0.089(1500 - y)+0.119y=225\)
Expand: \(133.5-0.089y + 0.119y=225\)
Combine like - terms: \(0.03y=225 - 133.5\)
\(0.03y=91.5\)
Solve for \(y\): \(y=\frac{91.5}{0.03}=3050\)

But this is wrong. Let's use the Pearson square method.
The difference between alfalfa hay (\(11.9\%\)) and required (\(15\%\)) is \(15 - 11.9=- 3.1\). The difference between corn (\(8.9\%\)) and required (\(15\%\)) is \(15 - 8.9 = 6.1\).
The ratio of corn to alfalfa hay is \(|15 - 11.9|:|15 - 8.9|=3.1:6.1\)
The total parts \(=3.1 + 6.1=9.2\)
Amount of corn \(x=\frac{6.1}{9.2}\times1500=\frac{6.1\times1500}{9.2}\approx994.57\) lb
Amount of alfalfa hay \(y=\frac{3.1}{9.2}\times1500=\frac{3.1\times1500}{9.2}\approx505.43\) lb

Answer:

The amount of corn is approximately \(994.57\) lb and the amount of alfalfa hay is approximately \(505.43\) lb.