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Chapter 3: Problem 59

For a particular salmon population, the relationship between the number \(S\) of spawners and the number \(R\) of offspring that survive to maturity is given by the formula $$R=\frac{4500 S}{S+500}$$ (a) Under what conditions is \(R>S ?\) (b) Find the number of spawners that would yield \(90 \%\) of the greatest possible number of offspring that survive to maturity. (c) Work part (b) with \(80 \%\) replacing \(90 \%\). (d) Compare the results for \(S\) and \(R\) (in terms of percentage increases) from parts (b) and (c).

Short Answer

Expert verified
(a) 0 < S < 4000; (b) S = 6750; (c) S = 9000; (d) 33.33% increase in S, 11.11% decrease in R.

Step by step solution

01

Identify the given formula

We are given the formula \( R = \frac{4500S}{S + 500} \) that relates the number of spawners \( S \) to the number of offspring \( R \) that survive to maturity.
02

Condition for R > S

We need to find when \( R > S \). Set up the inequality: \( \frac{4500S}{S + 500} > S \). Cross-multiply to get \( 4500S > S^2 + 500S \). Rearranging, we have \( S^2 - 4000S < 0 \). Solving \( S(S - 4000) < 0 \), we find the solution as \( 0 < S < 4000 \).
03

Find maximum R

Find the maximum \( R \) by taking the derivative of \( R(S) = \frac{4500S}{S+500} \) with respect to \( S \). Use the quotient rule to find \( R'(S) \) and set it to zero to find critical points. Upon solving, we find \( S = 4000 \) which gives the maximum \( R \). Substitute this back into the formula to get \( R_{max} = 3600 \).
04

Calculate 90% of maximum R

First calculate \( 90\% \) of \( R_{max} = 3600 \): \( 0.9 \times 3600 = 3240 \). We solve \( \frac{4500S}{S + 500} = 3240 \) for \( S \). Cross-multiplying and rearranging gives \( 4500S = 3240S + 1620000 \). Solving gives \( S = 6750 \).
05

Calculate 80% of maximum R

Now calculate \( 80\% \) of \( R_{max} = 3600 \): \( 0.8 \times 3600 = 2880 \). Solve \( \frac{4500S}{S + 500} = 2880 \) for \( S \). Cross-multiplying and rearranging gives \( 4500S = 2880S + 1440000 \). Solving gives \( S = 9000 \).
06

Percentage Increase Calculation

From part (b), we found \( S = 6750 \) for \( 90\% \) and \( S = 9000 \) for \( 80\% \). Calculate percentage increase in \( S \): \( \frac{9000 - 6750}{6750} \times 100\% \approx 33.33\% \). Calculate percentage increase in \( R = 3600 \) from \( 3240 \) to \( 2880 \): \( \frac{3240 - 2880}{3240} \times 100 = 11.11\% \).

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Key Concepts

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

Calculus
Calculus is a fundamental concept in precalculus problem solving, often used to analyze changes and determine values of interest. Calculus can be broadly divided into two branches: Differential Calculus and Integral Calculus. In this exercise, we utilize Differential Calculus to solve real-world problems.
  • Differential Calculus helps us find how a function changes as its input changes.
  • It involves finding "derivatives," which capture the rate of change of a function.
To identify maximum or minimum values, we find where the derivative is zero or undefined. In our problem, determining when the number of offspring is maximized involved finding a critical point using the derivative.
Inequalities
Inequalities are a mathematical way to express that two expressions are not equal, showing that one quantity is larger or smaller than the other. In this exercise, inequalities help determine under what conditions the number of offspring exceeds the number of spawners.
  • We set up the inequality \(\frac{4500S}{S + 500} > S\) to find where the offspring outnumber the spawners.
  • This leads us to the inequality \(S^2 - 4000S < 0\), which we solve to find the valid range for \(S\).
  • The solution \(0 < S < 4000\) tells us the interval where the number of offspring will be more than the number of spawners.
Understanding how to solve inequalities is essential in various mathematical and real-world applications.
Derivative
The derivative is a key tool in calculus used to measure how a function changes as its input changes. It's a representation of the function's rate of change or slope.
  • In this scenario, we used the derivative of the function \(R(S) = \frac{4500S}{S + 500}\) to find the maximum number of offspring.
  • Using the quotient rule, which is \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}\), we computed the derivative and set it equal to zero to find the critical points.
  • The critical point at \(S = 4000\) indicated where the function achieves a maximum, confirming that increasing the spawners up to this point maximizes the offspring.
Grasping how derivatives work allows us to understand and predict dynamic processes in physics, biology, and economics.
Optimization
Optimization involves finding the best solution or maximum/minimum value of a function under certain conditions. In this exercise, we explored optimizing the number of offspring.
  • To find the maximum number of offspring, we calculated the derivative \(R'(S)\) and found when it is zero.
  • The problem required finding the optimal number of spawners \(S\) that yields the greatest offspring, using calculus to locate this value.
  • We also calculated specific spawner amounts to achieve a certain percentage (like 90% and 80%) of the maximum offspring, involving algebraic manipulation of the given formula.
Optimization techniques are crucial in fields like business for cost reduction, resource management in engineering, and schedule optimization in logistics.
Percentage Calculation
Percentage calculations are used extensively to compare quantities relative to a whole, especially in determining changes or specifications as a fraction of a maximum.
  • In this exercise, the maximum possible offspring count needs to be adjusted to calculate for percentages like 90% and 80% of the total.
  • We calculated 90% of 3600 to be 3240, and solved \(\frac{4500S}{S + 500} = 3240\) to find \(S\).
  • The same process determined the spawner requirement for 80% as well, deriving specific values for practical decision-making.
  • Our understanding of percentages also helped find the percent increase in spawners and offspring numbers, providing insights into the efficiency of adjustments.
Learning to calculate percentages is a fundamental skill used across various disciplines for analysis, budgeting, and statistics.

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