A meteorologist finds that the temperature at the 6,000-foot level of a mountain is 76°F and the temperature at the 12,000-foot level of the mountain is 49°F. Write a linear equation to represent the temperature $T$ at an elevation of $x$, where $x$ is in thousands of feet.
We are given two data points relating the temperature \(T\) in degrees Fahrenheit to the elevation \(x\) in thousands of feet.
Our goal is to find a linear equation of the form \(T = mx + b\), where \(m\) is the slope and \(b\) is the vertical intercept.
The slope \(m\) of a line passing through two points \((x_1, T_1)\) and \((x_2, T_2)\) is calculated using the formula:
\[m = \frac{T_2 - T_1}{x_2 - x_1}\]
Substituting the known values \((x_1, T_1) = (6, 76)\) and \((x_2, T_2) = (12, 49)\):
\[m = \frac{49 - 76}{12 - 6} = \frac{-27}{6} = -4.5\]
The slope of the line is \(-4.5\). This indicates that for every 1000-foot increase in elevation, the temperature decreases by \(4.5^\circ\text{F}\).
To find the equation, we use the point-slope form:
\[T - T_1 = m(x - x_1)\]
Using the point \((6, 76)\) and the slope \(m = -4.5\):
\[T - 76 = -4.5(x - 6)\]
Now, we simplify to reach the slope-intercept form \(T = mx + b\):
\[T - 76 = -4.5x + (-4.5) \times (-6)\]
\[T - 76 = -4.5x + 27\]
Adding 76 to both sides:
\[T = -4.5x + 27 + 76\]
\[T = -4.5x + 103\]
Thus, the linear equation representing the temperature at an elevation \(x\) is \(T = -4.5x + 103\).
Linear relationships are frequently observed in meteorology and environmental science. For instance, if you are planning a mountain hike, knowing how the temperature drops with altitude helps you pack appropriate gear. By determining a linear model like the one solved here, you can accurately predict temperatures at various elevations, ensuring you stay prepared for the changing conditions of high-altitude environments.