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Chapter 4: Problem 10

Growth rate: An animal grows according to the formula $$ L=0.6 \log (2+5 T) . $$ Here \(L\) is the length in feet and \(T\) is the age in years. a. Draw a graph of length versus age. Include ages up to 20 years. b. Explain in practical terms what \(L(15)\) means, and then calculate that value. c. How old is the animal when it is 1 foot long? d. Explain in practical terms what the concavity of the graph means. e. Use a formula to express the age as a function of the length.

Short Answer

Expert verified
a. Plot graph: length vs age; b. \(L(15)\) is length at 15 years: compute \(L = 0.6 \log(77)\); c. Solve \(T\) for \(L = 1\); d. Concavity indicates growth rate changes; e. Age function: \(T = \frac{10^{\frac{L}{0.6}} - 2}{5}\).

Step by step solution

01

Preparing the graph components

The exercise requires plotting the graph of length versus age using the formula \( L=0.6 \log (2+5 T) \). We need age (\(T\)) values ranging from 0 to 20 years. We'll calculate the corresponding length (\(L\)) for each age to use as points for our graph.
02

Calculating the length for specific ages

Using the given function \(L=0.6 \log (2+5 T)\), compute the length for ages from 0 to 20. For example, \(T = 0\) gives \(L = 0.6 \log(2)\) and \(T = 20\) gives \(L = 0.6 \log(102)\). Creating a table of values will help in plotting the graph.
03

Plotting the graph

With the table of calculated values, plot the points on a graph with age \(T\) on the x-axis and length \(L\) on the y-axis. Connect the points smoothly to form the growth-rate curve from age 0 to 20 years.
04

Explaining and calculating \(L(15)\)

The value \(L(15)\) represents the length of the animal at 15 years of age. Compute \(L(15)\) using the formula: \(L(15) = 0.6 \log (2+5 \times 15) = 0.6 \log(77)\). Evaluate this expression to find \(L(15)\).
05

Finding age when \(L = 1\) foot

We need to find the age \(T\) when \(L = 1\). Set \(0.6 \log (2+5T) = 1\) and solve for \(T\). This yields \(\log (2+5T) = \frac{10}{6}\) or \(2+5T = 10^{\frac{10}{6}}\). Solve for \(T = \frac{10^{\frac{10}{6}} - 2}{5}\).
06

Understanding graph concavity

The concavity of the graph indicates how the growth rate changes with age. If the graph is concave up, then the growth rate increases. If concave down, the growth rate decreases. Use the second derivative test to determine concavity: \(L''(T)\) determines this characteristic.
07

Expressing age as a function of length

To express age \(T\) as a function of length \(L\), rearrange the formula: \(L = 0.6 \log (2+5T)\). Solving for \(T\) gives \(2 + 5T = 10^{\frac{L}{0.6}}\), so \(T = \frac{10^{\frac{L}{0.6}} - 2}{5}\). This is the inverse function expressing age \(T(L)\) in terms of \(L\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing
In this exercise, we are guided to create a graphical representation of how an animal grows over time using the function \( L=0.6 \log (2+5T) \). This function tells us the relationship between age \( T \) (in years) and length \( L \) (in feet). Graphing is crucial because it allows us to visually interpret and analyze the data.
To construct the graph, we:
  • Identify the domain of interest, which is age \( T \) from 0 to 20 years.
  • Calculate corresponding length \( L \) for each age within this range using our formula.
  • Plot these calculated points on a coordinate plane with age \( T \) on the x-axis and length \( L \) on the y-axis.
  • Connect the points to form a smooth curve.
This smooth curve represents the growth pattern of the animal over 20 years, showing us how lengths change with age. Plotting graphs aids in understanding the overall trend and behavior of the data.
Inverse Functions
Inverse functions flip the roles of inputs and outputs. In our context, while the original function \( L=0.6 \log (2+5T) \) gives us length based on age, its inverse will tell us the age based on a given length. This is particularly useful if we want to know how old the animal is when it measures a specific length.
To find the inverse function, we:
  • Start with the original formula: \( L = 0.6 \log (2+5T) \).
  • Solve for \( T \) in terms of \( L \).
  • Rearrange to get \( 2+5T = 10^{\frac{L}{0.6}} \).
  • Solve for \( T \): \( T = \frac{10^{\frac{L}{0.6}} - 2}{5} \).
This inverse function \( T(L) \) allows us to calculate the animal's age for any given length, showcasing a different perspective on the growth process.
Concavity
The concept of concavity informs us about the rate at which the growth of the animal is changing. Essentially, concavity tells us whether the growth is speeding up or slowing down.
Let's delve into what this means:
  • If the graph is concave up, this suggests that the growth is accelerating. In other words, the animal grows faster as it ages.
  • Conversely, if the graph is concave down, this indicates a decelerating growth rate, meaning the animal's growth is slowing down with age.
To determine concavity mathematically, we use the second derivative of the function, \( L''(T) \). The sign of this second derivative reveals the concavity:
  • A positive \( L''(T) \) implies that the graph is concave up.
  • A negative \( L''(T) \) implies that the graph is concave down.
Understanding the graph's concavity gives us insights into the shifting dynamics of growth as the animal ages.
Growth Rate
The growth rate in this context refers to how quickly or slowly the animal's length increases as it ages. This can be observed directly from the graph and understood through the function itself.
Here's how we analyze growth rates:
  • The slope of the tangent to the growth curve at any point signifies the instantaneous growth rate at that specific age.
  • In periods where the graph is steeper, the growth rate is higher, indicating rapid growth.
  • Where the graph flattens out, the growth rate slows down, indicating more gradual growth.
The function \( L=0.6 \log (2+5T) \) itself determines the growth rate pattern mathematically.
The focus on growth rates helps in predicting how significant changes in length will be, given a specific age interval.

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