Question

approximate linear models relate them. we measure the pressure ( p ) (kpa) and the boiling - point ( t ) of water. at a pressure of 100 kpa the boiling point of water is ( 100^{circ}c ) and drops by about ( 3.75^{circ}c ) for each 10 kpa drop in atmospheric pressure. find a linear function ( q ) that models the boiling - point ( t ) at an atmospheric pressure of ( x ) kilopascals.

Explanation:

Step1: Identify the slope - intercept form

The linear function is of the form $y = mx + b$, where $y$ is the boiling - point temperature, $x$ is the atmospheric pressure, $m$ is the slope, and $b$ is the y - intercept.

Step2: Calculate the slope $m$

We know that the boiling point drops by about $3.75^{\circ}C$ for each $10$ kPa drop in pressure. So, the slope $m=\frac{\Delta y}{\Delta x}=\frac{- 3.75}{10}=-0.375$.

Step3: Find the y - intercept $b$

When $x = 100$ kPa, $y = 100^{\circ}C$. Substitute into $y=mx + b$: $100=-0.375\times100 + b$.
Solve for $b$:
\[

$$\begin{align*} 100&=-37.5 + b\\ b&=100 + 37.5\\ b&=137.5 \end{align*}$$

\]

Step4: Write the linear function

The linear function $q(x)$ that models the boiling - point temperature $y$ in terms of the atmospheric pressure $x$ is $q(x)=-0.375x + 137.5$.

Answer:

$q(x)=-0.375x + 137.5$