Notes: At the beginning of 1998, the annual consumption of ice cream is in the U.S. was 12,582,000 pints and growing at a rate of 212,000,000 pints per...

Saturday, February 20, 2010

At the beginning of 1998, the annual consumption of ice cream is in the U.S. was 12,582,000 pints and growing at a rate of 212,000,000 pints per...

The data given with regards to the population of US in 1998 written in mathematical form:

Population: $\displaystyle{P}={268}{924}{000}$

Annual consumption of ice cream: $\displaystyle{C}={12}{582}{000}$

Rate of annual consumption of ice cream: $\displaystyle\frac{{{d}{C}}}{{\left.{d}{t}}}={212}{000}{000}$

Rate of population growth in the US in 1998: $\displaystyle\frac{{{d}{P}}}{{\left.{d}{t}}}={1856000}$

Since we are told that:

annual per capita growth = (annual consumption) / (population size)

We can use calculus, specifically the quotient rule to determine the annual per capita consumption of ice cream increasing at that time.

The formula is as follows:

$\displaystyle\frac{{d}}{{\left.{d}{x}}}{\left[\frac{{C}}{{P}}\right]}=\frac{{{P}\cdot{\left({d}\frac{{C}}{{\left.{d}{t}}}\right)}-{C}\cdot{\left({d}\frac{{P}}{{\left.{d}{t}}}\right)}}}{{{{P}}^{{2}}}}$

$\displaystyle\frac{{d}}{{\left.{d}{x}}}{\left[\frac{{C}}{{P}}\right]}=\frac{{{\left({268924000}\cdot{212}{000}{000}\right)}-{\left({12}{582}{000}\cdot{1}{856}{000}\right)}}}{{{{268924000}}^{{2}}}}$

$\displaystyle\frac{{d}}{{\left.{d}{x}}}{\left[\frac{{C}}{{P}}\right]}={0.788}$

The annual per capita consumption of ice cream increased at that time in 1998 was 0.788

Notes

Saturday, February 20, 2010

At the beginning of 1998, the annual consumption of ice cream is in the U.S. was 12,582,000 pints and growing at a rate of 212,000,000 pints per...

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