Trying to explain this to help my son

Question

 Based on data about the growth of a variety of ornamental trees, the following logarithmic model about these trees was determined:

               h(t) = 6.108 ln(t) + 6.099, where t = age of tree in years and h (t) = height of tree, in feet.    

                                                              (Note that "ln" refers to the natural log function) (explanation optional)

Using the model,

(a) At age 5 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?

(b) At age 15 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?

Answer 1

For part (a) you just plug in t=5 into the model and use a calculator to find the height of the tree.

h(5) = 6.108*ln(5) + 6.099 = 9.830 + 6.099 = 15.929 (rounded its 15.9 ft)

Similar for part (b), using t=15.

Logarithm (log) background:

The output of the log function "log( )" is an exponent. Also, "log" with no explicit base is often base 10, also called the common logarithm.

Example #1: log(100) = log₁₀(100) = 2 <--- The power to which the base (10 in this case) must be raised to get a value of 100, is 2: 10² = 100. [ NOTE: in some purely mathematical papers, "log" may be the natural log, be aware of context. ]

Natural logarithm, ln( ):

Example #2: ln(x) is the same as "log_e(x)" (not usually written this way, but using this for illustration)

The output of ln( ) is the power to which e (approx 2.71828183) must be raised to give you x.

ln(100) ≈ 4.60517. This means e^4.60517 = 100

ln() and e may seem strange, but they arise in the modeling of, and solutions to, many problems.

I encourage you to google "properties of logarithms." For example log_b(a*c) = log_b(a) + log_b(c) comes from a*c = b^log(a)_*b^log(c) = b^{(log(a)+log(c))} and taking log_b of both sides gives log_b(a*c) = log_b(a) + log_b(c).

Answer 2

Since you are given the equation for height vs. age of the tree, solving the problem is simply a matter of substituting the number of years into the equation. It would be harder if you were asked to determine how old the tree was when it was a given height.

The equation has the natural log (ln) because the older the tree, the slower it grows up (because it has to support all the rest of the tree at the same time)'

(a) at 5 years:

h(5) = 6.108 ln(5) + 6.099 = 15.9 ft

(b) at 15 years:

h(15) = 6.108 ln(15) + 6.099 = 22.6 ft

Using the equation for 10 years, get 20.2 ft - the tree grows 15.9 ft in the first 5 years, 4.3 ft in the second 5 years, and 2,4 ft in the third five years.

Answer 3

h(t) = 6.108*ln(t) + 6.099

At t=1, h(t) = 6.108*ln(1) + 6.099 = 6.099 ft. Note that ln(1) = 0

a) let t=5. ln(5) = 1.609. Plug this into the model to het h(5) = 6.108*1.609 + 6.099 = 15.9 ft.

b) let t=15. ln(15) = 2.708. Plug this into the model to get h(15) = 6.108*2.708 + 6.099 = 22.6 ft.