Question

answer
the expected weight of a catfish
that is 1 inch long.
the rate of change in the
expected length of the catfish
over its weight.
the expected length of a catfish
that weighs 1 pound.
the rate of change in the
expected weight of the catfish
over its length

Explanation:

Step1: Recall the concept of slope in a linear relationship

In a linear model \(y = mx + b\) (where \(y\) is the dependent variable, \(x\) is the independent variable, \(m\) is the slope and \(b\) is the y - intercept), the slope \(m=\frac{\Delta y}{\Delta x}\). In the context of the catfish length (\(x\)) and weight (\(y\)) graph, the slope represents the change in the dependent variable (weight) with respect to the change in the independent variable (length).

Step2: Analyze each option

• Option 1: The expected weight of a catfish that is 1 inch long would be the value of \(y\) when \(x = 1\). This is the y - intercept (\(b\)) of the line \(y=mx + b\) (not the slope).
• Option 2: The expected length of a catfish that weighs 1 pound would involve solving \(x\) for \(y = 1\) in the equation \(y=mx + b\), which is not the slope.
• Option 3: The rate of change in the expected length of the catfish over its weight would be \(\frac{\Delta x}{\Delta y}\), which is the reciprocal of the slope of the given line (since the given line has \(y\) (weight) as a function of \(x\) (length)).
• Option 4: The rate of change in the expected weight of the catfish over its length is \(\frac{\Delta y}{\Delta x}\), which is the slope of the line \(y\) (weight) vs \(x\) (length)

Answer:

The rate of change in the expected weight of the catfish over its length.