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Chapter 4: Problem 38

BUSINESS: Quality Control A company finds that the proportion of its light bulbs that will burn continuously for longer than \(t\) weeks is \(e^{-0.01 t}\). Find the proportion of bulbs that burn for longer than 10 weeks.

Short Answer

Expert verified
The proportion is approximately 0.9048.

Step by step solution

01

Understanding the Problem

We are given a function that describes the proportion of light bulbs burning longer than a certain time \( t \) weeks, which is \( e^{-0.01 t} \). We need to find the specific proportion for \( t = 10 \) weeks.
02

Substituting the Given Value

Substitute \( t = 10 \) into the function \( e^{-0.01 t} \). This gives us \( e^{-0.01 imes 10} \).
03

Calculating the Exponent

Calculate the exponent. This is \( -0.01 imes 10 = -0.1 \). So the expression becomes \( e^{-0.1} \).
04

Evaluating the Exponential Function

Evaluate \( e^{-0.1} \). Using a calculator, you find \( e^{-0.1} \approx 0.9048 \).
05

Conclusion

The proportion of bulbs that burn longer than 10 weeks is approximately 0.9048.

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Key Concepts

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

Quality Control
Quality control is crucial for businesses that manufacture products, especially those that must meet safety and performance standards. In the context of light bulbs, quality control involves ensuring that the bulbs last for a predictable amount of time. This is where statistical models, like exponential functions, come into play. Businesses use these models to predict product lifespans and maintain quality standards.

Control processes may include:
  • Monitoring the manufacturing process to prevent defects.
  • Testing samples from production to ensure consistency in performance.
  • Using predictive models to determine product lifespan and customer satisfaction.
By adhering to quality control measures, companies can increase user confidence and reduce costs associated with returns and repairs.
Business Applications
Understanding business applications of mathematical concepts such as exponential functions can greatly enhance decision-making processes. Many businesses use these formulas to optimize their operations, especially in product manufacturing.

Exponential functions are used in:
  • Predicting product failure rates, which helps in inventory management.
  • Pricing strategies, where the value of time-saving and longevity is considered in the product price.
  • Customer service planning, ensuring resources are efficiently used to address product issues.
These applications help companies to strategize and allocate resources wisely, ultimately leading to higher profitability.
Light Bulb Lifespan
The lifespan of light bulbs can be modeled using exponential functions, which describe how quickly bulbs are expected to fail over time. This specific mathematical model, expressed as \( e^{-0.01 t} \), illustrates that as time increases, the proportion of bulbs lasting decreases exponentially.

When we calculate \( e^{-0.1} \), it shows that approximately 90.48% of bulbs will last longer than 10 weeks. This kind of information is vital for businesses to:
  • Predict warranty periods and service life expectations.
  • Design marketing strategies around product longevity.
  • Anticipate customer needs for replacements or upgrades.
Understanding these intervals helps in planning production and guiding consumers on product usage.

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Most popular questions from this chapter

Since the development of the iPod, the stock price of Apple has been growing rapidly and has been approximately \(11 e^{0.34 x}\), where \(x\) is the number of years since 2000 (for \(0 \leq x \leq 10\) ). Find the relative growth rate of Apple's stock price at any time during that period.

If the national debt of a country (in trillions of dollars) \(t\) years from now is given by the indicated function, find the relative rate of change of the debt 10 years from now. $$ N(t)=0.5+1.1 e^{0.01 t} $$

Temperature A mug of beer chilled to 40 degrees, if left in a 70 -degree room, will warm to a temperature of \(T(t)=70-30 e^{-3.5 t}\) degrees in \(t\) hours. a. Find \(T(0.25)\) and \(T^{\prime}(0.25)\) and interpret your answers. b. Find \(T(1)\) and \(T^{\prime}(1)\) and interpret your answers.

Recall that the concentration of a drug in the bloodstream after \(t\) hours is \(c e^{-k t}\), where \(k\) is called the "absorbtion constant." If one drug has a larger absorbtion constant than another, will it require more or less time between doses? (Assume that both drugs have the same value of \(c\).)

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 \(S(p)=a p^{n}\), where \(a\) and \(n\) are positive constants.
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