Q. 76 - Calculus

Question

Suppose the height of a growing tree t years after it is planted is $h\left(t\right) = 0.25t^2+1$feet, as shown earlier at the right.

(a) Approximate the average height of the tree during the first five years of its growth, using a sample size of n = 5 times and then n = 10 times.

(b) Find the exact average height of the tree during the first five years of its growth.

(c) What was the average rate of growth of the tree over the first five years?

Step-by-Step Solution

Answer

(a) The average height at n=5 is 3.75 feet and at n=10 is 3.406 feet.

(b) The exact average height is 3.083 feet.

(c) The average rate of growth is 2.25 feet per year.

Step 1. Given Information.

$h\left(t\right) = 0.25t^2+1.$

And, a=0 , b=5.

Step 2. part (a) Formula for approximate height.

$$\begin{gathered} Formula to calculate height is: \\ \frac{1}{b-a}\sum _{k=1}^{n}f\left(x_k\right)∆x, where ∆x =\frac{b-a}{n}, and x_k=a+k∆x. \end{gathered}$$

Step 3. part(a) Solving for n=5.

$$\begin{gathered} ∆x = \frac{5-0}{5} = 1. \\ {x}_k=0+k*1 = k. \\ The average height, \\ =\frac{1}{b-a}\sum _{k=1}^{n}f\left(x_k\right)∆x \\ =\frac{1}{5-0}\sum _{k=1}^n\left(0.25k^2+1\right)\left(1\right) \\ =\frac{1}{5}\left(0.25\sum _{k=1}^n{k}^2 + \sum _{k=1}^{n}1\right) \\ =\frac{1}{5}\left(0.25\frac{n(n+1)(2n+1)}{6}+n\right) \\ Now n=5, \\ =\frac{1}{5}\left(0.25\frac{5(5+1)(10+1)}{6}+5\right) \\ =3.75 \\ Therefore, the approximate average height is 3.75 feet. \end{gathered}$$

Step 4. part(a) Solving for n=10.

$$\begin{gathered} ∆x = \frac{5-0}{10} = \frac{1}{2}. \\ {x}_k=0+\frac{k*1}{2} = \frac{k}{2}. \\ The average height, \\ =\frac{1}{b-a}\sum _{k=1}^{n}f\left(x_k\right)∆x \\ =\frac{1}{5-0}\sum _{k=1}^n\left(0.25\frac{k^2}{4}+1\right)\left(\frac{1}{2}\right) \\ =\frac{1}{10}\left(0.25\sum _{k=1}^n\frac{k^2}{4} + \sum _{k=1}^{n}1\right) \\ =\frac{1}{10}\left(\frac{0.25}{4}\times \frac{n(n+1)(2n+1)}{6}+n\right) \\ Now n=10, \\ =\frac{1}{10}\left(\frac{0.25}{4}\times \frac{10(10+1)(20+1)}{6}+10\right) \\ =3.406 \\ Therefore, the approximate average height is 3.406 feet. \end{gathered}$$

Step 5. part (b) The exact average height.

$$\begin{gathered} The exact average height will be: \\ =\frac{1}{5-0}{\int }_0^5\left(0.25t^2+1\right)dt \\ =\frac{1}{5}{\left[\frac{0.25{\left(t\right)}^3}{3}+t\right]}_0^5 \\ =\frac{1}{5}(15.417) = 3.083 \\ Therefore, the exact average height is 3.083 feet. \end{gathered}$$

Step 6. part (c) The average rate of growth.

$$\begin{gathered} The average rate of growth will be: \\ =\frac{h\left(5\right)-h\left(0\right)}{5-0} \\ = \frac{0.25{\left(5\right)}^2+1 -(0+1)}{5} \\ = 2.25 \\ Therefore, the average rate of growth over the first five years will be 2.25 feet per year. \end{gathered}$$