Solve each problem. See Example 9. Depth and flow. When the...

Solve each problem. See Example 9. Depth and flow. When the depth of the water in the Tangipahoa River at Robert, Louisiana, is 9.14 feet, the flow is 1230 cubic feet per second ( $\mathrm{ft}^{3} / \mathrm{sec}$ ). When the depth is 7.84 feet, the flow is $826 \mathrm{ft}^{3} / \mathrm{sec} .$ (U.S. Geological Survey, www.usgs.gov). Let $w$ represent the flow in cubic feet per second and $d$ represent the depth in feet. a) Write the equation of the line through $(9.14,1230)$ and $(7.84,826)$ and express $w$ in terms of $d .$ Round to two decimal places. b) What is the flow when the depth is $8.25 \mathrm{ft} ?$ c) Is the flow increasing or decreasing as the depth increases?

Solve each problem. See Example 9.
Depth and flow. When the depth of the water in the Tangipahoa River at Robert, Louisiana, is 9.14 feet, the flow is 1230 cubic feet per second ( $\mathrm{ft}^{3} / \mathrm{sec}$ ). When the depth is 7.84 feet, the flow is $826 \mathrm{ft}^{3} / \mathrm{sec} .$ (U.S. Geological Survey, www.usgs.gov). Let $w$ represent the flow in cubic feet per second and $d$ represent the depth in feet.
a) Write the equation of the line through $(9.14,1230)$ and $(7.84,826)$ and express $w$ in terms of $d .$ Round to two decimal places.
b) What is the flow when the depth is $8.25 \mathrm{ft} ?$
c) Is the flow increasing or decreasing as the depth increases?

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Key Concepts

Linear Equation

A linear equation represents a constant relationship between two variables, characterized by a straight line when plotted on a graph. This concept is fundamental when modeling real-world situations where one quantity changes at a constant rate relative to another.

Calculating Slope

The slope is the measure of the rate of change in the dependent variable relative to a change in the independent variable. It is computed as the difference in the dependent values divided by the difference in the independent values between two points, and it indicates whether the relationship is increasing or decreasing.

Point-Slope Form

The point-slope form is a method of writing the equation of a line when you know one point on the line and the slope. This form is particularly useful in setting up a linear model from empirical data.

Prediction Using a Linear Model

Once a linear equation is established, it can be used to predict the value of one variable based on the value of another. This is valuable for interpolation or extrapolation, helping to understand and forecast trends in various applications.

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