/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Find a function given by a formu... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 21

Find a function given by a formula in one of your textbooks for another class or some other handy source. If the formula involves more than one variable, assign reasonable values for all the variables except one so that your formula involves only one variable. Now make a graph, using an appropriate horizontal and vertical span so that the graph shows some interesting aspect of the function, such as a trend, significant values, or concavity. Carefully describe the function, formula, variables (including units), and graph, and explain how the graph is useful.

Short Answer

Expert verified
The function \( A = \pi r^2 \) describes the area of a circle based on its radius. The graph shows a parabolic curve with area increasing as radius increases, highlighting sensitivity to changes in radius.

Step by step solution

01

Select a function

Let's choose the function of the area of a circle, given by the formula: \( A = \pi r^2 \). Here, \( A \) represents the area, and \( r \) represents the radius of the circle.
02

Define variables

We'll focus on the function where the area \( A \) is dependent on the radius \( r \). For simplicity, we will keep \( \pi \) as its constant value, approximately 3.14. The variable \( r \) will be varied to observe changes in \( A \).
03

Graph the function

To make the graph, plot \( A \) on the vertical axis and \( r \) on the horizontal axis. Choose a suitable range for \( r \), such as from 0 to 10, and calculate \( A \) for each value of \( r \). The graph will depict a parabolic curve, starting at the origin and opening upwards.
04

Describe the graph

The graph shows a parabolic trend where the area \( A \) increases quadratically as \( r \) increases. It highlights how even a small increase in radius can lead to a significant increase in area, showing the sensitivity of area concerning changes in radius.
05

Explain the graph's usefulness

The graph clearly illustrates the relationship between the radius and area, useful in understanding scaling in geometric contexts. It highlights how a change in radius proportionally changes the area and can be critical in fields needing spatial optimization.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Graphing functions
When graphing functions, it's essential to understand how to represent a mathematical relationship visually. The function we examined is the area of a circle, which is expressed by the formula \( A = \pi r^2 \). Here, \( A \) is plotted alongside its dependent variable \( r \), which is the radius of the circle.

As you translate this to a graph, think of \( r \) as being on the horizontal axis while \( A \) occupies the vertical axis. Select a sensible range for \( r \), such as from 0 to 10, to explore meaningful sections of the curve. As you compute the values of \( A \) using various values of \( r \), you'll observe the characteristics of a parabolic curve opening upwards. This depiction underscores a key mathematical concept: the quadratic growth of the area with respect to the radius. It's the consistent upward trajectory that signifies a direct correlation between the increase in the radius and the exponential increase in area.

By graphing functions, students gain insight into how subtle changes in one variable can significantly impact the output, offering a valuable learning experience in interpreting and understanding data visually.
Variables in equations
Variables in equations are the building blocks for mathematical models. In the context of the circle's area, the variable \( r \) represents the radius, while \( A \) signifies the area. Here, \( \pi \) is a constant, approximately valued at 3.14, showcasing its unique role in geographical computations.

By fixing \( \pi \) and allowing \( r \) to vary, you manage a single-variable equation that is more straightforward to graph. The manipulation of \( r \) provides a dynamic insight into how variables work within an equation. Observing the changes in area with respect to fluctuating values of \( r \) highlights how different variables scale within mathematical relationships.

Incorporating variables in equations enables you to model real-world scenarios. For instance, if you are planning to build a circular garden, understanding how varying the radius affects the area helps effectively utilize space and resources. Thus, appreciating the role of variables adds depth to mathematical problem-solving and real-life applications.
Understanding mathematical models
Understanding mathematical models begins with recognizing how functions serve as representations of relationships in the real world. A mathematical model takes complex systems and simplifies them into comprehensible equations that predict or describe behavior.

With the function \( A = \pi r^2 \), you're using a mathematical model to illustrate how area changes in a circular system. The benefit of models is their ability to forecast or probe different scenarios, enabling informed decisions and strategies. For instance, by manipulating our model of the circle's area, you can predict the materials needed for construction concerning scale or assess environmental impacts, like how larger plots may affect land use.

This understanding provides a practical framework to quantify, analyze, and visualize phenomena. Whether in academia, engineering, or environmental studies, mathematical models offer valuable insights that improve planning, decision-making, and problem-solving processes.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The background for this exercise can be found in Exercises 11, 12, 13, and \(14 \mathrm{in} \mathrm{Sec}\) tion 1.4. A manufacturer of widgets has fixed costs of \(\$ 700\) per month, and the variable cost is \(\$ 65\) per thousand widgets (so it costs \(\$ 65\) to produce 1 thousand widgets). Let \(N\) be the number, in thousands, of widgets produced in a month. a. Find a formula for the manufacturer's total cost \(C\) as a function of \(N\). b. The highest price \(p\), in dollars per thousand widgets, at which \(N\) can be sold is given by the formula \(p=75-0.02 \mathrm{~N}\). Using this, find a formula for the total revenue \(R\) as a function of \(N\). c. Use your answers to parts a and \(b\) to find \(a\) formula for the profit \(P\) of this manufacturer as a function of \(N\). d. Use your formula from part c to determine the two break-even points for this manufacturer. Assume that the manufacturer can produce at most 500 thousand widgets in a month.

The amount of growth of plants in an ungrazed pasture is a function of the amount of plant biomass already present and the amount of rainfall. \({ }^{18}\) For a pasture in the arid zone of Australia, the formula $$ \begin{aligned} Y=&-55.12-0.01535 N \\ &-0.00056 N^{2}+3.946 R \end{aligned} $$ gives an approximation of the growth. Here \(R\) is the amount of rainfall (in millimeters) over a 3-month period, \(N\) is the plant biomass (in kilograms per hectare) at the beginning of that period, and \(Y\) is the growth (in kilograms per hectare) of the biomass over that period. (For comparison, 100 millimeters is about \(3.9\) inches, and 100 kilograms per hectare is about 89 pounds per acre.) a. Solve Equation (2.6) for \(R\). b. Ecologists are interested in the relationship between the amount of rainfall and the initial plant biomass if there is to be no plant growth over the period. Put \(Y=0\) in the equation you found in part a to get a formula for \(R\) in terms of \(N\) that describes this relationship. c. Use the formula you found in part b to make a graph of \(R\) versus \(N\) (again with \(Y=0\) ). Include values of \(N\) from 0 to 800 kilograms per hectare. This graph is called the isocline for zero growth. It shows the amount of rainfall needed over the 3 -month period just to maintain a given initial plant biomass. d. With regard to the isocline for zero growth that you found in part c, what happens to \(R\) as \(N\) increases? Explain your answer in practical terms. e. How much rainfall is needed just to maintain the initial plant biomass if that biomass is 400 kilograms per hectare? f. A point below the zero isocline graph corresponds to having less rainfall than is needed to sustain the given initial plant biomass, and in this situation the plants will die back. A point above the zero isocline graph corresponds to having more rainfall than is needed to sustain the given initial plant biomass, and in this situation the plants will grow. If the initial plant biomass is 500 kilograms per hectare and there are \(40 \mathrm{mil}-\) limeters of rain, what will happen to the plant biomass over the period?

A child has 64 blocks that are 1 -inch cubes. She wants to arrange the blocks into a solid rectangle \(h\) blocks long and \(w\) blocks wide. There is a relationship between \(h\) and \(w\) that is determined by the restriction that all 64 blocks must go into the rectangle. A rectangle \(h\) blocks long and \(w\) blocks wide uses a total of \(h \times w\) blocks. Thus \(h w=64\). Applying some elementary algebra, we get the relationship we need: $$ w=\frac{64}{h} . $$ a. Use a formula to express the perimeter \(P\) in terms of \(h\) and \(w\). b. Using Equation (2.3), find a formula that expresses the perimeter \(P\) in terms of the height only. c. How should the child arrange the blocks if she wants the perimeter to be the smallest possible? d. Do parts \(b\) and \(c\) again, this time assuming that the child has 60 blocks rather than 64 blocks. In this situation the relationship between \(h\) and \(w\) is \(w=60 / h\). (Note: Be careful when you do part c. The child will not cut the blocks into pieces!)

The quantity \(S\) of barley, in billions of bushels, that barley suppliers in a certain country are willing to produce in a year and offer for sale at a price \(P\), in dollars per bushel, is determined by the relation $$ P=1.9 S-0.7 $$ The quantity \(D\) of barley, in billions of bushels, that barley consumers are willing to purchase in a year at price \(P\) is determined by the relation $$ P=2.8-0.6 D . $$ The equilibrium price is the price at which the quantity supplied is the same as the quantity demanded. Find the equilibrium price for barley.

When a skydiver jumps from an airplane, her downward velocity, in feet per second, before she opens her parachute is given by \(v=176\left(1-0.834^{t}\right)\), where \(t\) is the number of seconds that have elapsed since she jumped from the airplane. We found earlier that the terminal velocity for the skydiver is 176 feet per second. How long does it take to reach \(90 \%\) of terminal velocity?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.