Chapter 3.2, Problem 98E

Arboriculture

The growth of a red oak tree is approximated by the function

$$G=–0.003t^3+0.137t^2+0.458t–0.839,2\le t\le 34$$

where $$G$$ is the height of the tree (in feet) and $$t$$ is its age (in years).

(a) Use a graphing utility to graph the function.

(b) Estimate the age of the tree when it is growing most rapidly. This point is called the point of diminishing returns because the increase in size will be less with each additional year.

(c) Using calculus, the point of diminishing returns can be found by finding the vertex of the parabola

$$y=–0.009t^2+0.274t+\mathrm{0.458.}$$

Find the vertex of this parabola.

(d) Compare your results from parts (b) and (c).