The height of a plant over time is shown in the table below. Using a logarithmic model, what is the approximate age of the plant when it is 19 inches tall?
| t, time in months | h, height in inches |
|---|---|
| 1 | 18 |
| 2 | 18.21 |
| 3 | 18.33 |
| 4 | 18.42 |
| 5 | 18.48 |
| 6 | 18.54 |
A. 10 months
B. 14 months
C. 16 months
D. 28 months
Answer (2)
Fit a logarithmic model to the data: h = a ln ( t ) + b .
Use the points (1, 18) and (2, 18.21) to find a = l n ( 2 ) 0.21 and b = 18 .
Set h = 19 and solve for t : 19 = l n ( 2 ) 0.21 ln ( t ) + 18 .
Calculate t = e 0.21 l n ( 2 ) ≈ 27.13 , so the closest answer is 28 months.
Explanation
Understanding the Problem We are given a table of plant heights over time and asked to find the age of the plant when it is 19 inches tall, using a logarithmic model. The data points are (1, 18), (2, 18.21), (3, 18.33), (4, 18.42), (5, 18.48), (6, 18.54).
Setting up the Logarithmic Model We will fit a logarithmic model of the form h = a ln ( t ) + b to the given data points. We can use two data points to solve for the coefficients a and b . Let's use the points (1, 18) and (2, 18.21).
Finding b Using the point (1, 18), we have 18 = a ln ( 1 ) + b . Since ln ( 1 ) = 0 , we get b = 18 .
Finding a Using the point (2, 18.21), we have 18.21 = a ln ( 2 ) + b . Substituting b = 18 , we get 18.21 = a ln ( 2 ) + 18 . Thus, a ln ( 2 ) = 0.21 , so a = l n ( 2 ) 0.21 .
The Logarithmic Model Our logarithmic model is h = l n ( 2 ) 0.21 ln ( t ) + 18 .
Solving for t We want to find the time t when the height h is 19 inches. So, we set h = 19 and solve for t : 19 = l n ( 2 ) 0.21 ln ( t ) + 18 .
Isolating the Logarithm Subtracting 18 from both sides, we get 1 = l n ( 2 ) 0.21 ln ( t ) .
Finding ln(t) Multiplying both sides by 0.21 l n ( 2 ) , we get ln ( t ) = 0.21 l n ( 2 ) .
Finding t To solve for t , we take the exponential of both sides: t = e 0.21 l n ( 2 ) .
Calculating t Calculating the value of t , we find that t ≈ 27.13 .
Final Answer The closest answer from the given options is 28 months.
Examples
Logarithmic models are useful in various real-world scenarios, such as modeling population growth, radioactive decay, and the spread of diseases. In environmental science, they can be used to model the concentration of pollutants in a lake over time. In finance, logarithmic models can describe the growth of investments or the depreciation of assets. Understanding logarithmic models helps us make predictions and informed decisions in these areas.
The approximate age of the plant when it is 19 inches tall is 28 months. This was determined by fitting a logarithmic model to the data and solving for time. The calculations showed that the height reaches 19 inches at around 27.13 months, rounding to the nearest option gives 28 months.
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