Growth of a red oak tree: finding point of dimishing returns

[jwpaine]

Hi. I have yet to take a calculus class, but here is our problem set from Algebra III which I need help figuring out....

The graph of a red oak tree is approximated by the function: G = -0.003(t)^3 + 0.137(t)^2 + 0.458t - 0.839

Where G is the height of the tree (in feet) and t (2 <= t <= 34) is it's age (in years)

Use a graphing utility to graph the function: Extimate 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.

Using calculus, the point of diminishing returns can also be found by finding the vertex of the parabola given by y = -0.009(t)^3 + 0.274(t) + 0.458. Find the vertex of this parabola.

Compare your results with your graphical estimate

So, she must have differentiated -0.003(t)^3 + 0.137(t)^2 + 0.458t - 0.839 down to -0.009(t)^2 + 0.274(t) + 0.458

If I find the vertex by completing the square (-b/(2a)) I get something close to X = 15.22, and substituting that into the trinomial I get Y = 4.49

I don't understand how this vertex shows me the age that the tree is growing most rapidly.

Thanks in advance for reading this and helping me understand what I'm actually doing.

John.

 

[skeeter]

the rate of change of the tree's growth is greatest when the sign of the derivative changes from (+) to (-).

the parabola is a graphical representation of the rate of growth ... the slope of the parabola changes from (+) to (-) at the vertex.

 

[jwpaine]

Thanks.

Now at which point (red dot or blue dot) would be my estimation.... at the blue dot I illustrated, right?

Now I dont see how this vertex I found of the derivative matches either dot of on the graphed cubic polynomial.

EDIT... OH.. I get it.... it would be around X = 15 because that appears to have the greatest change????

 

[skeeter]

the slope of the cubic (the rate of change of the tree's growth) is greatest between the red and blue dots.

 

[jwpaine]

Thank you, I understand now.

If I were to take the curve between the red and blue dot, and bend it into a parabola that matched the equation of the derivative, its vertex would be my max before it started depleting!!

Argh...can't wait for Calculus.

EDIT: Would you mind reading my report and seeing if I got everything right?
http://jpaine.net/math/problemset.pdf

Thanks,
John.