When a scientist dives in salt water to a depth of 9 feet below the surface, the pressure due to the atmosphere and surrounding water is 18.7 pounds per square inch. As the scientist descends, the pressure increases linearly. At a depth of 14 feet, the pressure is 20.9 pounds per square inch. If the pressure increases at a constant rate as the scientist's depth below the surface increases, which of the following linear models best describes the pressure $ p $ in pounds per square inch at a depth of $ d $ feet below the surface?

A. $ p = 0.44d + 0.77 $

B. $ p = 0.44d + 14.74 $

C. $ p = 2.2d - 1.1 $

D. $ p = 2.2d - 9.9 $

The given problem states that the pressure increases linearly as the scientist descends into the saltwater. This implies a constant rate of change in pressure with respect to depth. We can use the two provided data points to determine the linear relationship. The first data point is (9, 18.7) and the second data point is (14, 20.9).

For the linear model, we need to determine the slope [tex]($m$)[/tex] and the y-intercept [tex]($b$)[/tex] of the equation [tex]$p = md + b$.[/tex] The slope represents the rate of change of pressure with depth, and the y-intercept corresponds to the initial pressure at zero depth.

Using the two data points:

[tex]Slope ($m$) = $\frac{{20.9 - 18.7}}{{14 - 9}} = 0.44$[/tex]

Substitute the slope into one of the data points to find the y-intercept [tex]($b$)[/tex]:

[tex]$18.7 = 0.44 \times 9 + b$[/tex]

[tex]$b = 14.74$[/tex]

So the linear model that describes the pressure [tex]$p$[/tex] in pounds per square inch at a depth [tex]$d$[/tex] feet below the surface is [tex]$p = 0.44d + 14.74$.[/tex]

Therefore,the correct option is (b).

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