Question

examine the data below for a stalk of corn.

\\(\

$$\begin{array}{|c|c|c|c|c|}\\hline \\text{day, } x & 9 & 12 & 22 & 40 \\\\ \\hline \\text{height, } y \\text{ (in)} & 5 & 17 & 45 & 60 \\\\ \\hline\\end{array}$$

\\)

use logarithmic regression to find an equation of the form \\(y = a + b \ln(x)\\) to model the data.

\\(a = \\)
\\(b = \\)

using the graphing calculator, estimate to the nearest day how long it will take the corn to reach 6 feet tall:
\\(\text{days}\\)

Explanation:

Identify the given model and values

Using the Logarithmic Regression knowledge point
The logarithmic regression equation modeling the height of the corn stalk is:

$$ y = a + b \ln(x) $$

From the provided correct regression parameters in the image:

$$
LATEXBLOCK0
$$

Thus, the model is:

$$ y = -76.204 + 37.673 \ln(x) $$

where \(x\) represents the number of days, and \(y\) represents the height in inches.

Convert the target height to inches

The target height is given as \(6\text{ feet}\). Since the model's output \(y\) is in inches, we convert feet to inches:

$$ y = 6 \times 12 = 72\text{ inches} $$

Solve for the number of days

Using the Solving Logarithmic Equations knowledge point

$$
LATEXBLOCK1
$$

Round to the nearest day

Rounding \(51.11\) to the nearest whole number gives:

$$ x \approx 51\text{ days} $$

Answer:

Using the graphing calculator, estimate to the nearest day how long it will take the corn to reach 6 feet tall: <blank>51</blank> days