/*! 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 32 Per capita cigarette production ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 32

Per capita cigarette production in the United States during recent decades is approximately given by \(f(x)=-0.6 x^{2}+12 x+945\), where \(x\) is the number of years after \(1980 .\) Find the year when per capita cigarette production was at its greatest.

Short Answer

Expert verified
1990

Step by step solution

01

Identify the Mathematical Problem

To find the year when per capita cigarette production was at its greatest, we need to find the maximum value of the quadratic function \(f(x) = -0.6x^2 + 12x + 945\). This requires finding the vertex of the parabola, as it is a downward-opening parabola (the coefficient of \(x^2\) is negative).
02

Find the Vertex of the Parabola

The vertex of a parabola given by \(ax^2 + bx + c\) is found using the formula \(x = -\frac{b}{2a}\). In this function, \(a = -0.6\) and \(b = 12\). Plug these values into the formula to find \(x\).
03

Calculate \(x\) to Determine the Vertex

Using the formula from Step 2: \(x = -\frac{12}{2(-0.6)} = \frac{12}{1.2} = 10\). This means the maximum occurs 10 years after 1980.
04

Convert \(x\) to the Actual Year

Since \(x\) represents the number of years after 1980, we add 10 to 1980. Thus, the year when per capita cigarette production was at its greatest is 1990.

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.

Vertex of a Parabola
In a quadratic function, the vertex of a parabola is a crucial point, especially when determining maximum or minimum values. It is the point where the parabola changes direction. For a quadratic equation in the form of \(ax^2 + bx + c\), the vertex can be calculated using the formula \(x = -\frac{b}{2a}\). This formula helps us find the horizontal coordinate of the vertex.
Here are the steps to find the vertex:
  • Identify the coefficients: for \(f(x) = ax^2 + bx + c\), the coefficients are \(a\), \(b\), and \(c\).
  • Plug the coefficients into the vertex formula: \(x = -\frac{b}{2a}\).
  • The result gives you the \(x\)-value of the vertex. If needed, substitute \(x\) back into the function to get the \(y\)-value.
Knowing the vertex helps us understand the parabola's highest or lowest point. For example, in the given exercise, the parabola opens downwards, indicating that the vertex represents the maximum point of the function.
Maximum Value
The maximum value of a quadratic function like \(f(x) = -0.6x^2 + 12x + 945\) occurs at its vertex, given that the parabola opens downwards. The downward orientation is indicated by the negative coefficient of \(x^2\).
To find the maximum value:
  • Calculate the \(x\)-value of the vertex using \(x = -\frac{b}{2a}\). In this case, it gives \(x = 10\).
  • Substitute this \(x\)-value back into the quadratic function to get the \(y\)-value, which represents the maximum per capita production value.
In practical terms, the vertex's \(y\)-coordinate tells us how many cigarettes were produced per person at this maximum. It is not just a number but a snapshot of significant production at a specific time.
Per Capita Production
Per capita production refers to the total output divided by the population size, essentially giving the average production per person. In the context of the exercise, it tells us how much cigarette production was on average for each person in the U.S. during a particular year.
Understanding such data is important:
  • It shows consumption trends over time.
  • It helps evaluate the impact of economic, and public health policies on cigarette production.
  • A peak in per capita production, as computed from the vertex of the parabola, suggests the year when cigarette availability per person was highest.
This measure allows policymakers and researchers to gauge lifestyle changes and the effects of regulations intended to curb smoking.

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

Use implicit differentiation to find \(d y / d x\). \(x^{2} y=8\)

\(X\) and \(y\) are functions of \(t\) Differentiate with respect to \(t\) to find a relation between \(d x / d t\) and \(d y / d t\). \(2 x^{3}-5 x y=14\)

If a company spends \(r\) million dollars on research, its sales will be s million dollars, where \(r\) and \(s\) are related by \(s^{2}=r^{3}-55\) a. Find \(d s / d r\) by implicit differentiation and evaluate it at \(r=4, \quad s=3 .\) [Hint: Differentiate the equation with respect to \(r\).] b. Find \(d r / d s\) by implicit differentiation and evaluate it at \(r=4, s=3 .\) [Hint: Differentiate the original equation with respect to \(s\).] c. Interpret your answers to parts (a) and (b) as rates of change.

A rocket fired straight up is being tracked by a radar station 3 miles from the launching pad. If the rocket is traveling at 2 miles per second, how fast is the distance between the rocket and the tracking station changing at the moment when the rocket is 4 miles up? [Hint: The distance \(D\) in the illustration satisfies \(D^{2}=9+y^{2}\). To find the value of \(D\), solve \(\left.D^{2}=9+4^{2} .\right]\)

The equation \(x^{2}+4 y^{2}=100\) describes an ellipse. a. Use implicit differentiation to find its slope at the points \((8,3)\) and \((8,-3)\). b. Solve the equation for \(y\), obtaining two functions, and differentiate both to find the slopes at \(x=8 .\) [Answers should agree with part (a).] c. Use a graphing calculator to graph the two functions found in part (b) on an appropriate window. Then use NDERIV to find the derivatives at (or near) \(x=8 .\) [Answers should agree with parts (a) and (b).]
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

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.