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Chapter 4: Problem 26

After \(t\) days of advertisements for a new laundry detergent, the proportion of shoppers in a town who have seen the ads is \(1-e^{-0.03 t}\). How long must the ads run to reach: \(99 \%\) of the shoppers?

Short Answer

Expert verified
The ads must run for 154 days to reach 99% of the shoppers.

Step by step solution

01

Understand the Problem Statement

The function describing the proportion of shoppers who have seen the ads is \(1-e^{-0.03t}\). We need to find the number of days \(t\) such that 99% (or 0.99) of the shoppers have seen the ad, i.e., \(1-e^{-0.03t} = 0.99\).
02

Set up the Equation

Set the expression for the proportion equal to 0.99: \(1-e^{-0.03t} = 0.99\).
03

Solve for the Exponential Term

Subtract 1 from both sides of the equation: \(-e^{-0.03t} = 0.99 - 1\). This simplifies to \(-e^{-0.03t} = -0.01\).
04

Eliminate the Negative Sign

Divide both sides by -1: \(e^{-0.03t} = 0.01\).
05

Apply the Natural Logarithm

Take the natural logarithm of both sides to solve for \(t\): \(-0.03t = \ln(0.01)\).
06

Solve for t

Divide both sides by -0.03 to isolate \(t\): \[t = \frac{\ln(0.01)}{-0.03}\].
07

Calculate the Value of t

Calculate \(\ln(0.01)\) which is approximately -4.6052. Then substitute back to find \(t\): \[t \approx \frac{-4.6052}{-0.03} = 153.51\]. Since \(t\) must be a whole number, we'll round up to 154 days.

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Key Concepts

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

Natural Logarithm
The natural logarithm, often denoted as \( \ln(x) \), is a mathematical function that is crucial when dealing with exponential equations, especially in the context of growth and decay processes. It is based on the constant \( e \), approximately equal to 2.71828, which is the base of natural logarithms. When you see \( \ln(x) \), you are essentially asking, "To what power must \( e \) be raised to produce \( x \)?"
This concept is particularly important when solving equations involving exponential terms, such as \( e^{-0.03t} = 0.01 \) from the exercise.
  • Natural logarithms help to "undo" exponential functions, making it easier to solve for unknown variables.
  • By taking the natural logarithm of both sides of an equation, you can linearize an otherwise complex exponential relationship.
  • In our context, taking \( \ln(0.01) \) turns the equation from exponential into a simple linear form \(-0.03t = \ln(0.01)\).
Mathematical Modeling
Mathematical modeling involves creating a mathematical representation of a real-life scenario. This involves using equations, functions, or systems of equations to simulate how processes or phenomena might behave under different conditions.
In the given problem, the model used is \( 1 - e^{-0.03t} \), which represents the proportion of shoppers who have seen an advertisement over time \( t \).
This model assumes:
  • The number of people who see the advertisement increases over time
  • The growth rate decreases over time, a common feature in scenarios governed by exponential decay.
Mathematical models are powerful because:
  • They allow us to predict future outcomes based on current data, such as how long ads need to run to reach a certain percentage of viewers.
  • You can adjust model parameters to explore different scenarios, enhancing strategic decision-making.
Problem-Solving Process
Solving mathematical problems efficiently requires a structured process, particularly for complex scenarios involving exponential growth or decay. The problem-solving process we followed in the exercise can be outlined as follows:
  • **Understand the problem:** Clearly define what you're solving. In this case, determining how long it takes for 99% of shoppers to see the ad.
  • **Set up the equation:** Translate the problem into a mathematical equation, such as \( 1 - e^{-0.03t} = 0.99 \).
  • **Solve the equation:** Rearrange the equation step by step until you isolate the variable of interest (here, \( t \)).
  • **Use appropriate mathematical tools:** Apply tools such as the natural logarithm to simplify the equation.
  • **Calculate the solution:** Substitute any known values and perform the arithmetic to reach a numerical solution.
  • **Verify:** Double-check the solution to ensure it makes sense, especially when rounding is involved.
By following these steps, particularly the use of both qualitative assessments and quantitative tools, students can improve their accuracy and confidence in solving similar problems with exponential growth models.

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Most popular questions from this chapter

Use your graphing calculator to graph each function on a window that includes all relative extreme points and inflection points, and give the coordinates of these points (rounded to two decimal places). [Hint: Use NDERIV once or twice with ZERO.] (Answers may vary depending on the graphing window chosen.) $$ f(x)=\ln \left(1+x^{2}\right) $$

The following problems extend and augment the material presented in the text. a. Show that for a demand function of the form \(D(p)=c / p^{n}\), where \(c\) and \(n\) are positive constants, the elasticity is constant. b. What type of demand function has elasticity equal to 1 for every value of \(p\) ?

The number of people in a town of 50,000 who have heard an important news bulletin within \(t\) hours of its first broadcast is \(N(t)=\) \(50,000\left(1-e^{-0.4 t}\right) .\) Find the rate of change of the number of informed people: a. at time \(t=0\). b. after 8 hours.

If consumer demand for a commodity is given by the function below (where \(p\) is the selling price in dollars), find the price that maximizes consumer expenditure. $$ D(p)=8000 e^{-0.05 p} $$

A supply function \(S(p)\) gives the total amount of a product that producers are willing to supply at a given price \(p\). The elasticity of supply is defined as $$ E_{s}(p)=\frac{p \cdot S^{\prime}(p)}{S(p)} $$ Elasticity of supply measures the relative increase in supply resulting from a small relative increase in price. It is less useful than elasticity of demand, however, since it is not related to total revenue. Use the preceding formula to find the elasticity of supply for a supply function of the form \(S(p)=a e^{c p}\), where \(a\) and \(c\) are positive constants.
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