/*! 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 108 The Wohascum Center branch of Wo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 108

The Wohascum Center branch of Wohascum National Bank recently installed a digital time/temperature display which flashes back and forth between time, temperature in degrees Fahrenheit, and temperature in degrees Centigrade (Celsius). Recently one of the local college mathematics professors became concerned when she walked by the bank and saw readings of \(21^{\circ} \mathrm{C}\) and \(71^{\circ} \mathrm{F}\), especially since she had just taught her precocious five-year-old that same day to convert from degrees \(\mathrm{C}\) to degrees \(\mathrm{F}\) by multiplying by \(9 / 5\) and adding 32 (which yields \(21^{\circ} \mathrm{C}=69.8^{\circ} \mathrm{F}\), which should be rounded to \(70^{\circ} \mathrm{F}\) ). However, a bank officer explained that both readings were correct; the apparent error was due to the fact that the display device converts before rounding either Fahrenheit or Centigrade temperature to a whole number. (Thus, for example, \(21.4^{\circ} \mathrm{C}=70.52^{\circ} \mathrm{F}\).) Suppose that over the course of a week in summer, the temperatures measured are between \(15^{\circ} \mathrm{C}\) and \(25^{\circ} \mathrm{C}\) and that they are randomly and uniformly distributed over that interval. What is the probability that at any given time the display will appear to be in error for the reason above, that is, that the rounded value in degrees \(\mathrm{F}\) of the converted temperature is not the same as the value obtained by first rounding the temperature in degrees \(\mathrm{C}\), then converting to degrees \(\mathrm{F}\) and rounding once more?

Short Answer

Expert verified
There is a 5% probability that the display appears in error.

Step by step solution

01

- Understand the Conversion Formula

The conversion formula from degrees Celsius to degrees Fahrenheit is: \[ \text{F} = \left( \frac{9}{5} \times \text{C} \right) + 32 \]
02

- Define Rounding Conditions

We need to analyze the conditions where the rounded Fahrenheit value of the converted Celsius temperature is different when rounding Celsius first. Consider both scenarios:1. Rounding after conversion: \[ \text{Round} \left( \left( \frac{9}{5} \times \text{C} \right) + 32 \right) \]2. Converting after rounding Celsius:\[ \left( \frac{9}{5} \times \text{Round(C)} \right) + 32 \]
03

- Identify Error Conditions

The display is in error if: \[ \text{Round} \left( \left( \frac{9}{5} \times \text{C} \right) + 32 \right) eq \left( \left( \frac{9}{5} \times \text{Round(C)} \right) + 32 \right) \]
04

- Set Up Probability Calculations

The temperature ranges from \(15^{\circ} \text{C} \) to \(25^{\circ} \text{C} \). Define intervals where rounding could cause an error:
05

- Determine Critical Points

Identify intervals where rounding errors occur. For example, when \( \text{C} \) is close to 0.5, rounding can create issues. Examine intervals [15.0, 15.5), [15.5, 16.0), ..., [24.5, 25.0).
06

- Calculate Probability

Each interval where rounding matters has a width of 0.5掳C. There are 20 intervals: \[ [15.0, 15.5), [15.5, 16.0), ..., [24.5, 25.0) \]Since the temperature range is 10掳C (from 15掳C to 25掳C), the probability of falling into any one of these intervals is \( \frac{0.5}{10} = 0.05 \).

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.

Celsius to Fahrenheit Conversion
Understanding how to convert temperatures from Celsius to Fahrenheit is super useful, especially when you're traveling or dealing with scientific data. The formula for conversion is straightforward: multiply the Celsius temperature by \( \frac{9}{5} \) and then add 32. Here's how it looks: \[ \text{F} = \frac{9}{5} \times \text{C} + 32 \]

For example, if you have a temperature of 21掳C, you can convert it to Fahrenheit by calculating \[ \text{F} = \frac{9}{5} \times 21 + 32 = 69.8^{\text{F}} \]
This can be rounded to 70掳F, making it easier to read on a display or in a conversation.

This conversion is essential not just for daily life, but also in many academic and professional settings.
Rounding Errors
Rounding errors can cause confusion, especially in displays or precise calculations. A rounding error occurs when a number is approximated to a certain number of decimal places. For instance, when you convert 21掳C to Fahrenheit, you get 69.8掳F. Should you round it to 70掳F or display it as 69.8掳F?

In the example problem, the digital display first converts the temperature from Celsius to Fahrenheit, and then rounds the result. This can lead to small discrepancies. For instance, if the display shows 21掳C and converts it to 69.8掳F, rounding it to 70掳F afterward is straightforward. However, if the display rounds 21掳C to 21 before converting, it can yield a different Fahrenheit value, leading to apparent errors.

Understanding where these rounding discrepancies come from helps in making better sense of data and avoids confusion in real-world applications.
Probability Calculations
Probability is the likelihood of a certain event occurring. In this exercise, the goal is to determine the probability that the temperature display shows an error due to rounding discrepancies between Celsius and Fahrenheit.

To do this, consider the range of temperatures from 15掳C to 25掳C, which can be divided into smaller intervals where rounding might cause an error. Each interval, \[ [15.0, 15.5), [15.5, 16.0), ..., [24.5, 25.0) \], has a width of 0.5掳C. There are 20 such intervals within the 10掳C range.

The chance that the temperature falls within any one of these intervals is \[ \frac{0.5}{10} = 0.05 \] or 5%. This calculation shows that there is a 5% probability that rounding will result in a display error for temperatures randomly distributed between 15掳C and 25掳C. Knowing how to calculate such probabilities can be valuable in various domains, from weather forecasting to quality control in manufacturing.

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

A can is in the shape of a right circular cylinder of radius \(r\) and height \(h\). An intelligent ant is at a point on the edge of the top of the can (that is, on the circumference of the circular top) and wants to crawl to the point on the edge of the bottom of the can that is diametrically opposite to its starting point. As a function of \(r\) and \(h\), what is the minimum distance the ant must crawl?

Consider sequences of points in the plane that are obtained as follows: The first point of each sequence is the origin. The second point is reached from the first by moving one unit in any of the four "axis" directions (east, north, west, south). The third point is reached from the second by moving \(1 / 2\) unit in any of the four axis directions (but not necessarily in the same direction), and so on. Thus, each point is reached from the previous point by moving in any of the four axis directions, and each move is half the size of the previous move. We call a point approachable if it is the limit of some sequence of the above type. Describe the set of all approachable points in the plane. That is, find a necessary and sufficient condition for \((x, y)\) to be approachable.

Let \(g\) be a continuous function defined on the positive real numbers. Define a sequence \(\left(f_n\right)\) of functions as follows. Let \(f_0(x)=1\), and for \(n \geq 0\) and \(x>0\), let $$ f_{n+1}(x)=\int_1^x f_n(t) g(t) d t $$ Suppose that for all \(x>0, \sum_{n=0}^{\infty} f_n(x)=x\). Find the function \(g\). \(\quad\)

Can there be a multiplicative \(n \times n\) magic square \((n>1)\) with entries \(1,2, \ldots, n^2\) ? That is, does there exist an integer \(n>1\) for which the numbers \(1,2, \ldots, n^2\) can be placed in a square so that the product of all the numbers in any row or column is always the same?

a. Show that there is no cubic polynomial whose graph passes through the points \((0,0),(1,1)\), and \((2,16)\) and which is increasing for all \(x\). b. Show that there is a polynomial whose graph passes through \((0,0),(1,1)\), and \((2,16)\) and which is increasing for all \(x\). c. Show that if \(x_1
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

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.