Evaluating the growth of a tree with the calculation of an indefinite integral
Hello friends and followers of stem content, this time I want to explain how you can evaluate the growth of a tree knowing the growth rate when its height depends on time, for this we will see an example where the growth rate is exemplified by a differential equation, and to solve the problem we must integrate the differential equation to find the original growth function and make the respective substitutions, for this we propose the following example:
A nursery usually sells trees after 6 years of growth. The growth rate in those 6 years is given by the following differential equation:
dh / dt = 1.5t + 5
where t is the time in years.
h is the height in centimeters.
These trees at the time of planting measure 12 centimeters (t = 0).
a. Calculate its height after t years.
b. How tall are they at the time they are sold?
Understanding the problem
Since the rate of growth is given by a differential equation, to calculate the function describing that growth as a function of time we must find f(t) by integrating the differential equation, since the objective of integrating is to find the primitive of the differential.
To find the height of the tree at the time it is sold, this only occurs when t = 6 years have passed and we substitute it into f(t).
Solution to the problem
- First step: we find the function f(t) by integrating the differential equation dh / dt = 1.5t + 5 as shown below:
However, this function is not complete in all its terms, since at the moment of planting (t= 0) the trees measure 12 centimeters, so we have to add 12 to h(t):
- Second step: SWe substitute t = 6 years in h(t) , since we intend to calculate the height at the time the trees are sold and that happens when the trees reach 6 years of age:
Analysis of exercise responses
a. Calculate its height after t years.
This means that the height will vary as a function of time, therefore the function that we have to find is based on the original function that is found by integrating the differential equation, so the height of the tree as a function of time is:
b. How tall are they at the time they are sold?
The trees are sold after 6 years, and since the function describing the rate of growth over time is:
We simply substitute t = 6 years in h(t), which gives us a value of 69 centimeters, which means that from time zero to 6 years, the trees grew 57 centimeters (69 centimeters - 12 centimeters), which also means that they grew an average of 9.5 centimeters per year.
To prove that during the 6 years of tree growth, the trees grew to 69 centimeters, I will plot the parabola:
In the previous graph we can see that actually at t = 0 the trees had already grown 12 centimeters, and when they were 6 years old, which was when it was decided to sell them, they had already reached a size of 69 centimeters.
Recommended Bibliographic Reference
Calculus with Analytic Geometry. Volume I. 7th edition. Author: Larson and Hostetler.
Note: All equations in this post were prepared using Microsoft PowerPoint equation insertion tools. The graph of the parabola function was made using geogebra software.