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Chapter 2: Problem 18

The length \(L\), in inches, of a certain flatfish is given by the formula $$ L=15-19 \times 0.6^{t} \text {, } $$ and its weight \(W\), in pounds, is given by the formula $$ W=\left(1-1.3 \times 0.6^{t}\right)^{3} $$ Here \(t\) is the age of the fish, in years, and both formulas are valid from the age of 1 year. a. Make a graph of the length of the fish against its age, covering ages 1 to 8 . b. To what limiting length does the fish grow? At what age does it reach \(90 \%\) of this length? c. Make a graph of the weight of the fish against its age, covering ages 1 to 8 . d. To what limiting weight does the fish grow? At what age does it reach \(90 \%\) of this weight? e. One of the graphs you made in parts a and c should have an inflection point, whereas the other is always concave down. Identify which is which, and explain in practical terms what this means. Include in your explanation the approximate location of the inflection point.

Short Answer

Expert verified
Length reaches its limiting value of 15 inches by age 5, weight reaches 1 pound by age 4. Length graph is always concave down; weight graph has an inflection point around age 3.

Step by step solution

01

Part A: Graph the Length Against Age

To graph the length formula, \(L(t) = 15 - 19 \times 0.6^t\), calculate \(L\) for each year from \(t = 1\) to \(t = 8\). Find the values: \(L(1), L(2), ..., L(8)\), and plot them on a graph with \(t\) on the x-axis and \(L\) on the y-axis. Use appropriate scale and labels.
02

Part B: Calculate Limiting Length

The limiting length is the value of \(L(t)\) as \(t\) approaches infinity. As \(0.6^t\) approaches zero as \(t\) increases, the limiting length becomes \(L = 15\). Calculate \(90\%\) of \(L\), which is \(0.9 \times 15 = 13.5\). Find the smallest \(t\) such that \(L(t) \approx 13.5\). Iteratively calculate and find it happens around \(t \approx 5\).
03

Part C: Graph the Weight Against Age

For the weight formula \(W(t) = (1 - 1.3 \times 0.6^t)^3\), compute \(W\) for \(t = 1\) to \(t = 8\). Calculate each value: \(W(1), W(2), ..., W(8)\) and plot on a graph with \(t\) on the x-axis and \(W\) on the y-axis.
04

Part D: Calculate Limiting Weight

Like length, compute the limiting weight as \(t\) increases. As \(t\to \infty\), \(0.6^t\to 0\), making \(W = (1-0)^3 = 1\). Calculate \(90\%\) of \(W\), which is \(0.9\). Find the smallest \(t\) where \(W(t) \approx 0.9\). This occurs approximately at \(t \approx 4\).
05

Part E: Identify Graphs' Inflections

The length graph is concave down because \(L(t)\) reduces the effect of \(0.6^t\) over time. The weight graph has an inflection point because \(W(t) = (1-1.3 \times 0.6^t)^3\) changes concavity. The inflection is around \(t \approx 3\). The length graph suggests steady growth, while the weight graph's shift indicates acceleration in weight increase initially, then steadying.

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Key Concepts

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

Graphing Functions
Graphing functions is an essential skill in understanding mathematical patterns and relationships.
In this exercise, we are asked to plot two graphs: one for the length of a flatfish and another for its weight as a function of its age.
Both graphs, with age on the x-axis, demonstrate how these characteristics of the fish change over time, from the age of 1 to 8 years.
  • For the length, the formula is given by: \(L(t) = 15 - 19 \times 0.6^t \).
  • For the weight, the formula is \(W(t) = (1 - 1.3 \times 0.6^t)^3 \).
By graphing these equations, we visually explore how the characteristics of the fish evolve.
This practical approach helps make abstract functions tangible and shows how different formulas can model real-world phenomena.
The length graph decreases the influence of the exponent as the fish ages, showing how it approaches its maximum length limit.
The weight graph, on the other hand, shifts and reveals where the fish experiences different growth rates over time.
Graphing thus serves as a visual representation to analyze patterns, growth, and limits of the flatfish's development.
Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent.
In our problem, the term \(0.6^t\) exemplifies exponential decay since the base, 0.6, is less than 1, causing the term to shrink as \(t\) increases.
This type of function characterizes the flatfish's length and weight.
  • The length equation, \(L(t) = 15 - 19 \times 0.6^t\), illustrates how the influence of the exponential decay reduces over time, resulting in the growth rate of the fish slowing down as it nears its maximum length.
  • The weight equation, \(W(t) = (1 - 1.3 \times 0.6^t)^3\), conveys a similar idea where the exponential component dampens, eventually approaching a stable weight.
The beauty of exponential functions lies in modeling phenomena with decay or rapid change.
In biological contexts, these functions are used to simulate growth processes, radioactive decay, and more.
Understanding exponential decay helps us predict long-term behaviors of such systems, like the flatfish's size limit.
This concept reveals not only how systems evolve but also the implications of initial conditions and long-term projections.
Calculus Concepts
Calculus introduces powerful tools for studying how things change and enables the understanding of curves and their characteristics.
In the context of this exercise, two primary calculus concepts are of interest: limiting behavior and the presence of inflection points.
  • Limiting behavior: As \(t\) tends toward infinity in both the length and weight formulas, the exponential terms \(0.6^t\) reduce towards zero.
  • Therefore, the flatfish approaches a limiting length of 15 inches and a limiting weight of 1 pound.
The limiting behavior provides insight into the ultimate size and weight the flatfish aims to reach as it ages.
Inflection points are another key aspect highlighted in our solutions.
For the weight graph, a change in concavity occurs, hinting at different growth dynamics.
The inflection point illustrates where the rate of growth transitions from speeding up to slowing down.
This, intercepted mathematically, explains how biological organisms experience varying growth stages.
Ultimately, calculus allows us to quantify and visualize these changes, offering a comprehensive view of growth patterns in natural phenomena.

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Most popular questions from this chapter

The manager of an employee health plan for a firm has studied the balance \(B\), in millions of dollars, in the plan account as a function of \(t\), the number of years since the plan was instituted. He has determined that the account balance is given by the formula \(B=60+7 t-50 e^{0.1 t}\). a. Make a graph of \(B\) versus \(t\) over the first 7 years of the plan. b. At what time is the account balance at its maximum? c. What is the smallest value of the account balance over the first 7 years of the plan?

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