The following situation can be described by a linear function. Draw a graph of the function and use the graph to answer the given question. Be sure to identify the independent and dependent variables. Then briefly discuss whether a linear model is reasonable for the situation described. The diameter of a tree increases by 0.2 inch with each passing year. When the tree was first observed, its diameter was 5 inches. Estimate the time at which the tree started growing. The number of years is the variable. The diameter in inches is the variable.

Step 1 ：The problem describes a situation where the diameter of a tree increases by 0.2 inch with each passing year. When the tree was first observed, its diameter was 5 inches. We are asked to estimate the time at which the tree started growing.

Step 2 ：This situation can be described by a linear function, where the number of years is the independent variable and the diameter of the tree in inches is the dependent variable.

Step 3 ：We can set up a linear equation to represent this relationship. The diameter of the tree increases by 0.2 inch each year, so the slope of the line is 0.2. When the tree was first observed, its diameter was 5 inches, so the y-intercept is 5.

Step 4 ：The equation of the line is \(y = 0.2x + 5\), where y is the diameter of the tree and x is the number of years.

Step 5 ：To find the time at which the tree started growing, we need to find the x-intercept of the line, which is the value of x when y is 0.

Step 6 ：Setting y to 0 in the equation gives us \(0 = 0.2x + 5\). Solving for x gives us \(x = -25\).

Step 7 ：Since x represents the number of years, a negative value indicates that the tree started growing before it was first observed.

Step 8 ：Final Answer: The tree started growing \(\boxed{25}\) years before it was first observed.