/*! 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 1 The humidity is currently \(32 \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 1

The humidity is currently \(32 \%\) and falling at a rate of 4 percentage points per hour. a. Estimate the change in humidity over the next 20 minutes. b. Estimate the humidity 20 minutes from now.

Short Answer

Expert verified
Change: 1.33% decrease; Future humidity: 30.67%.

Step by step solution

01

Determine Rate of Change Per Minute

The humidity is decreasing at a rate of 4 percentage points per hour. First, convert this rate to a per-minute rate. There are 60 minutes in an hour, so the rate of change per minute is \( \frac{4}{60} = \frac{1}{15} \) percentage points per minute.
02

Calculate Change in Humidity for 20 Minutes

Now, determine the change in humidity over 20 minutes. Multiply the rate of change per minute by 20: \( 20 \times \frac{1}{15} = \frac{20}{15} = \frac{4}{3} \approx 1.33 \) percentage points.
03

Estimate Future Humidity

Subtract the change in humidity over 20 minutes from the current humidity to estimate the future humidity. The current humidity is 32%, and the change is approximately 1.33 percentage points, so the estimated future humidity is \( 32 - 1.33 = 30.67 \% \).

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.

Understanding Percentage Decrease
Percentage decrease occurs when the value of a quantity goes down over time. It's essential to recognize this as a way of understanding how much something lessens by comparing it to its original quantity. To find a percentage decrease:
  • First, identify the original percentage amount鈥攈ere, the humidity starts at 32%.
  • Determine how much it decreases. In this exercise, it's falling by a rate of 4 percentage points per hour.
  • Calculate the decrease as a percentage of the original value. However, in this case, we are directly given the decrease rate in percentage points.
Understanding percentage decrease helps us predict and measure changes more accurately.
Breaking Down Per-Minute Rate
The per-minute rate helps us understand changes happening continuously over time in smaller increments. In this exercise, the humidity is said to decrease by 4 percentage points every hour. Since we need changes per minute, we divide by 60 (minutes in one hour):
  • Calculate per-minute rate: \(\frac{4}{60} = \frac{1}{15} \) percentage points per minute.
This rate is crucial for short-term estimations, allowing us to apply it over different time spans simply and effectively. By knowing the per-minute rate, you can understand changes within shorter periods accurately.
Estimating Future Conditions
Estimating the future conditions involves predicting upcoming changes based on current data. In this case, you need to estimate the humidity 20 minutes in the future. To do this:
  • Use the per-minute rate from the previous step: \( \frac{1}{15} \) percentage points per minute.
  • Multiply the per-minute rate by the number of minutes鈥�20, to find the change within that time: \(20 \times \frac{1}{15} = \frac{4}{3} \approx 1.33\).
  • Subtract this change from the current humidity percentage of 32%.
This process effectively estimates a new condition鈥�30.67% humidity鈥攂y applying the calculated decrease over a specified time frame.
The Role of Mathematical Conversion
Mathematical conversion is about transforming numbers or units to better suit the context of a problem. In the exercise, we convert hours to minutes because we want to find how much humidity decreases per minute, rather than per hour. Here's how mathematical conversion works in this context:
  • Convert 4 percentage points per hour to per-minute rate: divide by 60 since 60 minutes constitute an hour.
This results in a unit that is more applicable for our purposes, especially because we are dealing with a 20-minute time span. Mathematical conversions ensure calculations remain sensible and proportionate to the problem's parameters.

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

For Activities 5 through \(10,\) given the units of measure for production and the units of measure for cost or revenue a. Write the units of measure for the indicated marginal. b. Write a sentence interpreting the marginal as an increase. Revenue is given by \(R(q)\) hundred dollars when \(q\) thousand units are sold; \(R^{\prime}(16)=3\).

a. Write the derivative formula. b. Locate any relative extreme points and identify the extreme as a maximum or minimum. $$ f(x)=12\left(1.5^{x}\right)+12\left(0.5^{x}\right) $$

For Activities 5 through \(10,\) given the units of measure for production and the units of measure for cost or revenue a. Write the units of measure for the indicated marginal. b. Write a sentence interpreting the marginal as an increase. Revenue is given by \(R(q)\) thousand dollars when \(q\) hundred units are sold; \(R^{\prime}(16)=0.15 .\)

River Flow Rate Suppose the flow rate (in cubic feet per second, \(\mathrm{cfs}\) ) of a river in the first 11 hours after the beginning of a severe thunderstorm can be modeled as \(s\) $$ f(h)=-0.865 h^{3}+12.05 h^{2}-8.95 h+123.02 \mathrm{cfs} $$ where \(h\) is the number of hours after the storm began. a. What are the flow rates for \(b=0\) and \(b=11\) ? b. Calculate the location and value of any relative extreme(s) for the flow rates on the interval between \(h=0\) and \(h=11\)

The apparent temperature \(A\) in degrees Fahrenheit is related to the actual temperature \(t^{\circ} \mathrm{F}\) and the humidity \(100 h \%\) by the equation \(A=2.70+0.885 t-78.7 h+1.20 t h \quad{ }^{\circ} \mathrm{F}\) (Source: W. Bosch and L. G. Cobb, "Temperature Humidity Indices," UMAP Module 691, UMAP Journal, vol. \(10,\) no. 3 , Fall \(1989,\) pp. \(237-256)\) a. If the humidity remains constant at \(53 \%\) and the actual temperature is increasing from \(80^{\circ} \mathrm{F}\) at a rate of \(2^{\circ} \mathrm{F}\) per hour, what is the apparent temperature and how quickly is it changing with respect to time? b. If the actual temperature remains constant at \(100^{\circ} \mathrm{F}\) and the relative humidity is \(30 \%\) but is dropping by 2 percentage points per hour, what is the apparent temperature and how quickly is it changing with respect to time?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.