A degree-day is a unit used to measure the fuel requirements...

A degree-day is a unit used to measure the fuel requirements of buildings. By definition, each degree that the average daily temperature is below $65^{\circ} \mathrm{F}$ is 1 degree-day. For example, an average daily temperature of $42^{\circ} \mathrm{F}$ constitutes 23 degree-days. If the average temperature on January 1 was $42^{\circ} \mathrm{F}$ and fell $2^{\circ} \mathrm{F}$ for each subsequent day up to and including January 10 how many degree-days are included from January 1 to January $10 ?$

Key Concepts

Temperature Threshold

The temperature threshold is a base or reference temperature used in energy consumption calculations to determine heating or cooling needs. In many contexts, this is the temperature at which no heating or cooling is required, and deviations from this threshold dictate the energy demand based on the degree to which temperatures differ from it.

Arithmetic Sequence Summation

An arithmetic sequence is a series of numbers in which each term increases or decreases by a constant value. When the daily temperature changes in a constant manner, the calculations for degree-days over several days can be approached as an arithmetic series, allowing for efficient summation of the daily differences relative to the temperature threshold.

Degree-Day

A degree-day is a unit used in energy analysis that measures the demand for heating or cooling based on the difference between a daily average temperature and a reference threshold. It is calculated by subtracting the average temperature (when it is below or above a designated base temperature) from the base temperature, with the resulting value indicating the energy needed to raise or lower the temperature to the base level.

Average Daily Temperature

The average daily temperature is the arithmetic mean of the temperatures recorded over the course of a day. It provides a simplified metric for the day’s thermal condition and is crucial for energy calculations, such as degree-days, as it represents the starting point for determining daily energy requirements.

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