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Chapter 2: Problem 18

In a study of a marine tubeworm, scientists developed a model for the time \(T\) (measured in years) required for the tubeworm to reach a length of \(L\) meters. \({ }^{23}\) On the basis of their model, they estimate that 170 to 250 years are required for the organism to reach a length of 2 meters, making this tubeworm the longest-lived noncolonial marine invertebrate known. The model is $$ T=\frac{20 e^{b L}-28}{b} $$ Here \(b\) is a constant that requires estimation. What is the value of \(b\) if the lower estimate of 170 years to reach a length of 2 meters is correct? What is the value of \(b\) if the upper estimate of 250 years to reach a length of 2 meters is correct?

Short Answer

Expert verified
For a length of 2 meters: \( b \approx 0.026 \) for 170 years, and \( b \approx 0.021 \) for 250 years.

Step by step solution

01

Understanding the Problem

We need to find the constant \( b \) in the equation \( T=\frac{20 e^{b L}-28}{b} \) when \( L = 2 \) and \( T \) is either 170 or 250 years.
02

Setting Up the Equation for the Lower Estimate

To find the value of \( b \) for the lower estimate, set \( T = 170 \) and \( L = 2 \) in the equation: \[ 170 = \frac{20 e^{2b} - 28}{b} \].
03

Solving for b in the Lower Estimate Equation

Rewrite the equation:\[ 170b = 20 e^{2b} - 28 \]. This equation is transcendental and doesn't have a simple algebraic solution, so numerical methods, such as using a graphing calculator, are needed to approximate \( b \).
04

Setting Up the Equation for the Upper Estimate

For the upper estimate, set \( T = 250 \) and \( L = 2 \):\[ 250 = \frac{20 e^{2b} - 28}{b} \].
05

Solving for b in the Upper Estimate Equation

Rewrite the equation:\[ 250b = 20 e^{2b} - 28 \].Similar to the lower estimate scenario, this is also a transcendental equation. Use numerical methods to approximate \( b \).
06

Approximation Using Numerical Methods

Use a calculator or software that supports numerical solutions to approximate \( b \). For the lower estimate, \( b \) is approximately \( 0.026 \), and for the upper estimate, \( b \) is approximately \( 0.021 \).

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Key Concepts

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

Transcendental Equations
A transcendental equation is one that includes one or more transcendental functions. These functions go beyond the capabilities of algebraic equations, as they include special functions such as exponential, logarithmic, or trigonometric functions. Unlike polynomial equations, transcendental equations do not have solutions that can be expressed using simple algebra.
In the exercise, we see a transcendental equation when we encounter the term involving the exponential function: \[ 170b = 20 e^{2b} - 28 \]
Here, the equation involves the variable in the exponent, making it transcendental. Such equations cannot be solved using traditional algebraic methods and often require numerical approaches for approximation.
Numerical Methods
Numerical methods are techniques used to approximate solutions to mathematical equations that do not have straightforward solutions. For transcendental equations, such as those found in the exercise, numerical methods come into play. These methods include:
  • Newton's method
  • Bisection method
  • Graphical methods using graphing calculators or software
Each of these approaches uses iterative procedures to converge upon the approximate value of the solution. In our exercise, because the equations involve complex transcendental components, using software or calculators that support such numerical computations is essential. Typically, these tools employ algorithms to iterate toward the solution by starting with a guess and refining it until the result meets a predefined criterion for accuracy.
Exponential Functions
An exponential function is a mathematical function of the form \( f(x) = a \,e^{bx} \), where \( e \) is the base of the natural logarithm, and \( a \) and \( b \) are constants. This type of function exhibits exponential growth or decay, making it powerful in modeling dynamic processes, like the growth of our marine tubeworm.

In the equation from the exercise, \( e^{bL} \) represents an exponential function that is crucial to the model鈥檚 ability to predict the timeline for a tubeworm to reach a specified length. Because exponential functions grow rapidly, they are often used to describe processes in nature that increase or decrease quickly over time. When combined with numerical methods, these functions can model the tubeworm's growth accurately, even when precise algebraic solutions aren鈥檛 feasible.

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Most popular questions from this chapter

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.

The monthly profit \(P\) for a widget producer is a function of the number \(n\) of widgets sold. The formula is $$ P=-15+10 n-0.2 n^{2} . $$ Here \(P\) is measured in thousands of dollars, \(n\) is measured in thousands of widgets, and the formula is valid up to a level of 15 thousand widgets sold. a. Make a graph of \(P\) versus \(n\). b. Calculate \(P(1)\) and explain in practical terms what your answer means. c. Is the graph concave up or concave down? Explain in practical terms what this means. d. The break-even point is the sales level at which the profit is 0 . Find the break-even point for this widget producer.

Here is a model for the number of students enrolled in U.S. public high schools as a function of time since 1965 : $$ N=-0.02 t^{2}+0.44 t+11.65 . $$ In this formula \(N\) is the enrollment in millions of students, \(t\) is the time in years since 1965 , and the model is applicable from 1965 to \(1985 .\) a. Calculate \(N(7)\) and explain in practical terms what it means. b. In what year was the enrollment the largest? What was the largest enrollment? c. Find the average yearly rate of change in enrollment from 1965 to 1985 . Is the result misleading, considering your answer to part b?

If you roll \(N\) dice, then the probability \(p=p(N)\) that you will get exactly 4 sixes is given by $$ p=\frac{N(N-1)(N-2)(N-3)}{24} \times\left(\frac{1}{6}\right)^{4}\left(\frac{5}{6}\right)^{N-4} $$ a. What is the probability, rounded to three decimal places, of getting exactly 4 sixes if 10 dice are rolled? How many times out of 1000 rolls would you expect this to happen? b. How many dice should be rolled so that the probability of getting exactly 4 sixes is the greatest?

In everyday experience, the measures of temperature most often used are Fahrenheit \(F\) and Celsius \(C\). Recall that the relationship between them is given by $$ F=1.8 C+32 $$ Physicists and chemists often use the Kelvin temperature scale. \({ }^{15}\) You can get kelvins \(K\) from degrees Celsius by using $$ K=C+273.15 . $$ a. Explain in practical terms what \(K\) (30) means, and then calculate that value. b. Find a formula expressing the temperature \(C\) in degrees Celsius as a function of the temperature \(K\) in kelvins. c. Find a formula expressing the temperature \(F\) in degrees Fahrenheit as a function of the temperature \(K\) in kelvins. d. What is the temperature in degrees Fahrenheit of an object that is 310 kelvins?
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