From 1970 to 1990 the average per person annual consumption of whole milk declined at a constant rate from 101 to 41 quarts per year. During the same period, the average annual consumption of low fat milk increased at a constant rate from 25 to 60 quarts per year. What was the average annual per person consumption when the consumption of whole milk and low fat milk was the same?Choose 1 answer:(A) 53 quarts(B) 55 quarts(C) 57 quarts(D) 59 quarts
Q. From 1970 to 1990 the average per person annual consumption of whole milk declined at a constant rate from 101 to 41 quarts per year. During the same period, the average annual consumption of low fat milk increased at a constant rate from 25 to 60 quarts per year. What was the average annual per person consumption when the consumption of whole milk and low fat milk was the same?Choose 1 answer:(A) 53 quarts(B) 55 quarts(C) 57 quarts(D) 59 quarts
Determine rate of decline for whole milk consumption: Determine the rate of decline for whole milk consumption.The decline in whole milk consumption over 20 years (from 1970 to 1990) is from 101 quarts to 41 quarts. This is a total decline of 101−41=60 quarts over 20 years.Rate of decline for whole milk $= \frac{\text{Total decline}}{\text{Number of years}} = \frac{\(60\) \text{ quarts}}{\(20\) \text{ years}} = \(3\) \text{ quarts per year}.
Determine rate of increase for low fat milk consumption: Determine the rate of increase for low fat milk consumption.\(\newline\)The increase in low fat milk consumption over the same \(20\) years is from \(25\) quarts to \(60\) quarts. This is a total increase of \(60 - 25 = 35\) quarts over \(20\) years.\(\newline\)Rate of increase for low fat milk = Total increase / Number of years = \(\frac{35 \text{ quarts}}{20 \text{ years}} = 1.75\) quarts per year.
Set up equation for finding year of equal milk consumption: Set up an equation to find the year when the consumption of whole milk and low fat milk was the same.\(\newline\)Let \(x\) be the number of years after \(1970\) when the consumptions are equal.\(\newline\)Starting consumption of whole milk + (Rate of decline \(* x\)) = Starting consumption of low fat milk + (Rate of increase \(* x\))\(\newline\)\(101 - 3x = 25 + 1.75x\)
Solve equation for x: Solve the equation for x.\(\newline\)Combine like terms:\(\newline\)\(101 - 25 = 3x + 1.75x\)\(\newline\)\(76 = 4.75x\)\(\newline\)Divide both sides by \(4.75\) to find \(x\):\(\newline\)\(x = \frac{76}{4.75}\)\(\newline\)\(x = 16\)
Calculate average annual per person consumption: Calculate the average annual per person consumption when the consumption of whole milk and low fat milk was the same.\(\newline\)We can use either the whole milk or low fat milk equation to find the consumption at year \(x\) (\(16\) years after \(1970\)).\(\newline\)Using the whole milk equation:\(\newline\)Consumption of whole milk \(= 101 - 3x\)\(\newline\)Consumption of whole milk \(= 101 - 3(16)\)\(\newline\)Consumption of whole milk \(= 101 - 48\)\(\newline\)Consumption of whole milk \(= 53\) quarts