The temperature-humidity index I (or humidex, for short) is...

The temperature-humidity index $I$ (or humidex, for short) is the perceived air temperature when the actual temperature is $T$ and the relative humidity is $h,$ so we can write $I=f(T, h) .$ The following table of values of $I$ is an excerpt from a table compiled by the National Oceanic \& Atmospheric Administration. (a) What is the value of $f(95,70) ?$ What is its meaning? (b) For what value of $h$ is $f(90, h)=100 ?$ (c) For what value of $T$ is $f(T, 50)=88 ?$ (d) What are the meanings of the functions $I=f(80, h)$ and $I=f(100, h) ?$ Compare the behavior of these two functions of $h .$

The temperature-humidity index $I$ (or humidex, for short) is the perceived air temperature when the actual temperature is $T$ and the relative humidity is $h,$ so we can write $I=f(T, h) .$ The following table of values of $I$ is an excerpt from a table compiled by the National Oceanic \& Atmospheric Administration.
(a) What is the value of $f(95,70) ?$ What is its meaning?
(b) For what value of $h$ is $f(90, h)=100 ?$
(c) For what value of $T$ is $f(T, 50)=88 ?$
(d) What are the meanings of the functions $I=f(80, h)$ and $I=f(100, h) ?$ Compare the behavior of these two functions of $h .$

Key Concepts

Temperature-Humidity Index

This concept refers to a measure that combines air temperature and relative humidity to estimate the perceived temperature by human bodies. It captures the effect of moisture in the air on the ability to cool oneself through evaporation, hence providing a more realistic feel of the weather conditions than the actual temperature alone.

Function of Two Variables

The problem employs a function f that takes two inputs—temperature (T) and relative humidity (h)—and maps them to a single output, the humidex (I). This perspective underscores the idea of functions in mathematics that depend simultaneously on multiple variables, each influencing the final outcome.

Interpreting Tabulated Meteorological Data

The use of a table to display values of the temperature-humidity index illustrates a common method in meteorology and other sciences for presenting precomputed values of complex functions. Understanding how to read and extract information from such tables is essential for making practical predictions and analyses in real-world applications.

Inverse Evaluation in Functions

Several parts of the question require determining an input value given a fixed output and one other input. This involves the concept of inverse function evaluation or solving for an independent variable when the function's output is known, which is a critical mathematical technique in problems where functional relationships do not have a simple algebraic inversion.

Parameter Sensitivity Analysis

The comparison between functions like I=f(80, h) and I=f(100, h) highlights how altering one parameter (in this case, temperature) affects the overall behavior of the function relative to the other parameter (humidity). This analysis is important in understanding the responsiveness and sensitivity of the perceived temperature to changes in environmental conditions.

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