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Chapter 4: Problem 1

a. Identify the output variable to be optimized and the input variable(s). b. Sketch and label a diagram. c. Write a model for the output variable in terms of one input variable. d. Answer the question posed. A rectangular-shaped garden has one side along the side of a house. The other three sides are to be enclosed with 60 feet of fencing. What is the largest possible area of such a garden?

Short Answer

Expert verified
The largest possible area is 450 square feet.

Step by step solution

01

Identify Variables

The output variable to be optimized is the area of the garden. The input variables are the dimensions of the garden, which we'll denote as length \( x \) for the side parallel to the house, and width \( y \) for the perpendicular side.
02

Sketch and Label Diagram

Draw a rectangle representing the garden with one side along the house. Label the length of the side parallel to the house as \( x \), and the side perpendicular to the house (enclosed by the fence) as \( y \). Indicate that three sides of this rectangle are enclosed by 60 feet of fencing.
03

Write Equation for Fencing Constraints

Since the garden's fencing is 60 feet and it covers three sides (two widths and one length), the equation is: \( x + 2y = 60 \).
04

Express Width in Terms of Length

Solve the constraint equation \( x + 2y = 60 \) for \( y \): \( y = \frac{60 - x}{2} \).
05

Write Area Function

The area \( A \) of the rectangle is given by \( A = x \times y \). Substituting \( y = \frac{60 - x}{2} \) gives: \( A(x) = x \times \frac{60 - x}{2} = 30x - \frac{x^2}{2} \).
06

Optimize the Area Function

To find the maximum area, we need to find the critical points of \( A(x) = 30x - \frac{x^2}{2} \). Take the derivative \( A'(x) = 30 - x \), and set it to zero: \( 30 - x = 0 \). Solve for \( x \) to get \( x = 30 \).
07

Verify Maximum Area

Using the second derivative test, \( A''(x) = -1 \) which is negative, indicating a maximum. Calculate \( y = \frac{60 - 30}{2} = 15 \). The maximum area is \( A = 30 \times 15 = 450 \) square feet.

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Key Concepts

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

Calculus
Calculus is a branch of mathematics that deals primarily with change and motion. It provides the tools needed to perform optimization problems, like finding maximums or minimums.
In this problem, calculus is used to maximize the area of a rectangular garden. By using derivatives, we can find critical points where the function has potential maximum or minimum values.
  • First, we find the derivative of the area function.
  • Second, set the derivative to zero to find critical points.
  • Then, use the second derivative test to confirm if it鈥檚 a maximum or minimum.
Understanding calculus allows us to systematically approach and solve optimization challenges in various fields.
Rectangular Garden Problem
The rectangular garden problem is a classic example of a practical optimization problem. Here, we want to maximize the area of a garden using a fixed amount of fencing.
This problem helps illustrate how to model a real-world situation mathematically and solve it efficiently.
The task requires recognizing the constraints鈥攈ere, the fencing available鈥攁nd applying them to derive an equation. This requires sketching the problem and labeling the dimensions to aid understanding.
By solving the related equations, we can express the dimensions that optimize the garden's area.
Maximizing Area
Maximizing area is the goal of this optimization problem. Given limited fencing, we aim to find dimensions that give the maximum garden area.
First, express one variable in terms of another using the fencing constraint equation.
  • Substitute this into the area formula.
  • This gives a function for the area as a single variable, making it easier to differentiate.
The critical point obtained by setting the derivative to zero indicates where the area is maximized. Verifying this with the second derivative test confirms it's indeed a maximum.
The result is a clear understanding of how to optimize space with constraints.
Mathematical Modeling
Mathematical modeling involves creating a mathematical representation of a real-world scenario. It helps with understanding and solving problems systematically.
In this garden problem, mathematical modeling starts with identifying variables and constraints. One key step is visually representing the problem with a diagram.
  • Establish relationships between variables鈥攗sing equations.
  • Translate physical constraints into mathematical terms.
This model guides us to find solutions effectively. By converting a word problem into mathematical language, you gain clarity in analyzing and deriving solutions.

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Most popular questions from this chapter

Write the first and second derivatives of the function and use the second derivative to determine inputs at which inflection points might exist. \(f(x)=98\left(1.2^{x}\right)+120\left(0.2^{x}\right)\)

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A softball diamond is a square with each side measuring 60 feet. Suppose a player is running from second base to third base at a rate of 22 feet per second. a. At what rate is the distance between the runner and home plate changing when the runner is halfway to third base? b. How far is the runner from home plate at this time?

Senior Population (Projected) The U.S. Bureau of the Census prediction for the percentage of the population 65 years and older can be modeled as $$ \begin{aligned} p(x)=&-0.00022 x^{3}+0.014 x^{2} \\ &-0.0033 x+12.236 \text { percent } \end{aligned} $$ where \(x\) is the number of years since \(2000,\) data from \(0 \leq x \leq 50\) (Source: Based on data from U.S. Census Bureau, National Population Projections, \(2008 .)\) a. Determine the year between 2000 and 2050 in which the percentage is predicted to be increasing most rapidly, the percentage at that time, and the rate of change of the percentage at that time. b. Repeat part \(a\) for the most rapid decrease.

Landfill Usage (Historic) The yearly amount of garbage (in million tons) taken to a landfill outside a city during selected years from 1980 through 2010 is given below. Landfill Usage: Annual Amount of Garbage Taken to a Landfill $$ \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Garbage } \\ \text { (million tons) } \end{array} \\ \hline 1980 & 81 \\ \hline 1985 & 99 \\ \hline 1990 & 115 \\ \hline 1995 & 122 \\ \hline 2000 & 132 \\ \hline 2005 & 145 \\ \hline 2010 & 180 \\ \hline \end{array} $$ a. Using the table values only, identify during which 5 year period the amount of garbage showed the slowest increase. What was the average rate of change during that 5 -year period? b. Write a model for the data. c. Locate the input of the point of slowest increase. How is this input located using the first derivative? How is this input located using the second derivative? d. In what year was the rate of change of the yearly amount of garbage the smallest? What was the rate of increase in that year?
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