/*! 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 84 The percentage of movie advertis... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 84

The percentage of movie advertising as a share of newspapers' total advertising revenue from 1995 to 2004 can be approximated by $$ p(t)=\left\\{\begin{array}{ll} -0.07 t+6.0 & \text { if } t \leq 4 \\ 0.3 t+17.0 & \text { if } t>4 \end{array}\right. $$ where \(t\) is time in years since \(1995 .^{15}\) a. Compute \(\lim _{t \rightarrow 4^{-}} p(t)\) and \(\lim _{t \rightarrow 4^{+}} p(t)\), and interpret each answer. HINT [See Example 4.] b. Is the function \(p\) continuous at \(t=4\) ? What does the answer tell you about newspaper revenues?

Short Answer

Expert verified
The left-handed limit at \(t = 4\) is 5.72%, while the right-handed limit at \(t = 4\) is 17.2%. This indicates that there was a sudden jump in the percentage of movie advertising as a share of newspapers' total advertising revenue from 5.72% just before the end of the 4th year to 17.2% just after the beginning of the 5th year. The function is not continuous at \(t = 4\), which means that this significant increase in the newspapers' movie advertising revenue occurred abruptly at the beginning of the 5th year.

Step by step solution

01

Identify Each Piece of the Function

The given function is divided into two parts: 1. p(t) = -0.07t + 6.0, if t <= 4 2. p(t) = 0.3t + 17.0, if t > 4
02

Calculate the Left-Handed Limit

To calculate the left-handed limit, i.e., the limit of p(t) as t approaches 4 from the left, we'll use the first part of the function: 1. p(t) = -0.07t + 6.0, if t <= 4 Now, we can take the limit as t approaches 4: \(\lim_{t \rightarrow 4^-} p(t) = \lim_{t \rightarrow 4} (-0.07t + 6.0)\) = -0.07(4) + 6.0 = 5.72
03

Calculate the Right-Handed Limit

To calculate the right-handed limit, i.e., the limit of p(t) as t approaches 4 from the right, we'll use the second part of the function: 2. p(t) = 0.3t + 17.0, if t > 4 Now, we can take the limit as t approaches 4: \(\lim_{t \rightarrow 4^+} p(t) = \lim_{t \rightarrow 4} (0.3t + 17.0)\) = 0.3(4) + 17 = 17.2
04

Interpret the Limits

The left-handed limit is 5.72, which represents the percentage of movie advertising as a share of newspapers' total advertising revenue just before the end of the 4th year or just before the beginning of the 5th year. Similarly, the right-handed limit is 17.2, which represents the percentage of movie advertising as a share of newspapers' total advertising revenue just after the beginning of the 5th year.
05

Check for Continuity and Interpret Result

To determine if the function is continuous at t=4, we need to check if both the left-handed limit and the right-handed limit exist and match: \(\lim_{t \rightarrow 4^-} p(t) = 5.72\) (left-handed limit) \(\lim_{t \rightarrow 4^+} p(t) = 17.2\) (right-handed limit) Since the left-handed limit and the right-handed limit are different, the function is not continuous at t=4. This means that there is a sudden jump in the percentage of movie advertising as a share of newspapers' total advertising revenue from the end of the 4th year to the beginning of the 5th year. The newspapers' movie advertising revenue after 4 years shows a significant increase (from 5.72% to 17.2%) at the beginning of the 5th year.

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.

Piecewise Functions
Piecewise functions are mathematical expressions defined by different formulas or rules for different intervals of the independent variable. In the problem at hand, we have a piecewise function that models the percentage of movie advertising as a share of newspapers' total advertising revenue. The function is defined over two intervals: from the beginning of 1995 to the end of 1999 (\(t \leq 4\)), and from the start of 2000 onwards (\(t > 4\)). For the first interval, the function is expressed as \(p(t) = -0.07t + 6.0\), and for the second, as \(p(t) = 0.3t + 17.0\). This structure captures different trends or behaviors in advertising revenue before and after a particular point in time. Understanding piecewise functions involves assessing each piece on its own and seeing how they fit together in the entire context, usually covering continuous or discrete changes at the interval boundaries.
Continuity
Continuity in calculus is a property of functions that indicates there are no sudden jumps or breaks in the curve of the graph. To determine if a function is continuous at a given point, we check that the limit of the function as it approaches the point from the left and right exists and equals the function's value at that point.In the given exercise, the function \(p(t)\) must be evaluated for continuity at \(t = 4\). We calculated that the left-handed limit is 5.72 and the right-handed limit is 17.2. Since these two values are not equal, the function has a jump discontinuity at \(t = 4\). This indicates that our piecewise function experiences an instantaneous change in the percentage at this point. In practical terms, for the newspaper's advertising revenue, it implies there was a considerable change in the advertising strategy or market conditions after 1999 that caused the percentage to jump at the beginning of the new interval.
Advertising Revenue Analysis
Analyzing advertising revenue, particularly with piecewise functions, helps businesses and researchers understand how different factors can affect revenue over time. In the exercise, the function \(p(t)\) models how movie advertising revenue changes as a portion of total newspaper advertising. Between 1995 and 2004, the revenue exhibits two distinct behaviors, captured by the piecewise segments.Understanding these trends can help in identifying periods of growth or decline. The sharp increase seen at \(t = 4\) suggests a significant change in advertising effectiveness or strategy. It points to an opportunity to analyze potential business decisions made at that time. This type of analysis is crucial for forecasting future trends, adjusting strategies, and making informed decisions to maximize revenue.

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

(Compare Exercise 29 of Section 10.4.) The percentage of mortgages issued in the United States that are subprime (normally classified as risky) can be approximated by $$ A(t)=\frac{15}{1+8.6(1.8)^{-t}} \quad(0 \leq t \leq 9) $$ where \(t\) is the number of years since the start of 2000 . a. Estimate \(A(6)\) and \(A^{\prime}(6)\). (Round answers to two significant digits.) What do the answers tell you about subprime mortgages? b. \(\mathrm{T}\) Graph the extrapolated function and its derivative for \(0 \leq t \leq 16\) and use your graphs to describe how the derivative behaves as \(t\) becomes large. (Express this behavior in terms of limits if you have studied the sections on limits.) What does this tell you about subprime mortgages? HINT [See Example 5.]

Compute the derivative function \(f^{\prime}(x)\) algebraically. (Notice that the functions are the same as those in Exercises \(1-14 .)\) HINT [See Examples 2 and \(3 .]\) $$ f(x)=2 x^{2}+x $$

Find the equation of the tangent to the graph at the indicated point. HINT [Compute the derivative algebraically; then see Example \(2(\mathrm{~b})\) in Section \(10.5 .]\) $$ f(x)=3 x+1 ; a=1 $$

If a stone is thrown down at 120 feet per second from a height of 1,000 feet, its height after \(t\) seconds is given by \(s=1,000-120 t-16 t^{2}\). Find its instantaneous velocity function and its velocity at time \(t=3\). HINT [See Example 4.]

Another prediction of Einstein's Special Theory of Relativity is that, to a stationary observer, clocks (as well as all biological processes) in a moving object appear to go more and more slowly as the speed of the object approaches that of light. If a spaceship travels at a speed of warp \(p\), the time it takes for an onboard clock to register one second, as measured by a stationary observer, will be given by $$ T(p)=\frac{1}{\sqrt{1-p^{2}}} \text { seconds } $$ with domain \([0,1)\). Estimate \(T(0.95)\) and \(T^{\prime}(0.95)\). What do these figures tell you?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.