Calculus Applications of Integrationrates using integration ### Calculus and Integrals: Real-World Applications### Problem 4: Rate of Fans Entering a Stadium**Problem Statement:**Football fans enter a sports stadium at a rate of $ r(t) $ people per hour, where $ t $ is time in hours. If there are 18,000 people in the stadium at the end of the first hour, write an expression that would represent the number of fans inside the stadium by the end of the second hour.**Solution Approach:**To find the number of fans inside the stadium by the end of the second hour, integrate the rate function $ r(t) $ over the time interval from 1 hour to 2 hours. Add this to the initial number of fans (18,000) already in the stadium at the end of the first hour.### Problem 2: Water Treatment Plant Processing Rate**Problem Statement:**A water treatment plant processes gray water at a rate given by $ r(t) = 16e^{0.2t} $ million gallons per year, where $ t $ is the time measured in years for $ 0 \leq t \leq 5 $. Write an expression that gives the amount of gray water processed by the treatment plant during the time interval $ 0 \leq t \leq 5 $.**Solution Approach:**To determine the total amount of gray water processed over the given time interval, integrate the rate function $ r(t) $ from $ t = 0 $ to $ t = 5 $ years. This will provide the total volume of gray water processed by the plant during this period.---#### Detailed Explanation of Diagrams and Graphs:In both problem statements above, consider plotting the rate functions $ r(t) $ against time to visualize the functional change over the given intervals. For Problem 2 specifically:- **Graph of $ r(t) = 16e^{0.2t} $** - **Y-axis:** Rate of gray water processing in million gallons per year. - **X-axis:** Time $ t $ in years. - **Curve:** Exponential curve starting from $ r(0) = 16 $ at $ t = 0 $ and increasing as $ t $ grows, reflecting the exponential increase in the rate of water processing.Such visualizations can aid in understanding how the amount processed

Since you have posted a multiple question according to guildlines I will solve first (Q2)…

2. A water treatment plant processes gray water at a rate given by r(t) = 16e0.2 million gallons per year, where t is the time measured in years for 0 St≤ 5. Write an expression that gives the amount of gray water processed by the treatment plant during the time interval 0 ≤ t ≤5.

2. A water treatment plant processes gray water at a rate given by r(t) = 16e0.2 million gallons per year, where t is the time measured in years for 0 St≤ 5. Write an expression that gives the amount of gray water processed by the treatment plant during the time interval 0 ≤ t ≤5.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: A miniature quadcopter flies in a curve line from one point to another. Suppose that the path of the…

A: We have to find arc length L= ∫ab1+dydx2dx

Q: Techniques of Integration 1 4 a lnx Find the value of constant (a) if J, dx =5. %3D

Q: Prove the function is convex function T f(x, y) = log (ex+ey) where x, y € RT E

A: fx,y=logex+ey

Q: dx | = xVx² - 1

Q: ntegral Calculus - Plane Area 1. Find the area in the first quadrant enclosed by y = (x2 + 4)/x2, x…

A: Topic = Area

Q: Which component is missing from the biologist's process? A biologist wonders if the life span of…

A: Assignment: Things Biologist done correctly Used proper labelling of peices of paper Sufficient…

Q: Expanding cube The edges of a cube increase at a rate of 2 cm>s. How fast is the volume changing…

A: Given, The edges of a cube increase at a rate of 2cm/s. Find How fast the volume changing when…

Q: Subject: calculus Using integration, find the area of the region enclosed by the lines 2x+y-6=0…

A: Given equation of lines are as following 2x+y-6=0 ; 3x-2y-1=0; x-3y+4=0 To find the area enclosed by…

Q: Find the point on the line -3x + 5y U which is closest to the point (1,3).

A: Introduction: For distance to be shortest between point and the line, there must be line…

Q: Use integration (t + 7)dt

Q: Substitution after rewriting the integrand Evaluate V4 = dx. - x6

A: Here f(x)=x24-x6 We have to integrate the function by substitutionI=∫x24-x6 dx substitute…

Q: S √x XHI хр

A: the solution is given in step2

Q: V-9+16* d.x 16*

Q: Reverse the order of integration and evaluate the double integral dxdy

Q: Changing order of integration Write the integral ∫20 ∫10 ∫1-y0 dz dy dx in the five other possible…

A: Consider the provided integral,

Q: Calculus 11th Edition - Ron Larson Chapter 4.5 - Integration by Substitution Finding the Area of a…

Q: NTEGRATE THE FOLLOWING (use integration techniques) - S sin rsin4rdx

Q: Find the area, in square units, bounded above by f(x) = x²+2x+1 and g(x) = 2x + 10 and bounded below…

A: First we identify the region which we need to find we have f(x)=x2+2x+1, g(x)=2x+10 area is above…

Q: Evaluate and simplify the integral +x+3 ad +6xª +9 °

A: This is a problem of integration. Based on the general formula of partial fraction decomposition we…

Q: Find the area of the surface generated by revolving about the x-axis the ellipse x = acost, y =…

A: We need to find area. NOTE - As per Bartleby's guidelines we can solve one question from multiple…

Q: maximize: subject to: 2x1 + x2 0≤x≤ 5 0 ≤ x2 ≤7 x1 + x₂ < 90

A: (.) Given , In simplex method we add slack variables.Let slack variables, then inequalities…

Q: Given the curve y = x' and the line y = 4x in quadrant 1 Find the abscissa of the centroid X of the…

A: We have to find abscissa of centroid of region bounded by the given curve and line.

Q: = " L+3)dzdrd0. Then the region of integration is presented in the figure the figure the figure the…

A: Here we will find the correct figure.

Q: Evaluate I cose cosec² (20) cot (20) do.

A: As per the question we are given an indefinite Integra which we have to evaluate.

Q: Gauss Integration 3 2 Evaluate f Saxy(2 -x2) (2- xy) dx dy over the region shown.

A: I=∫∫Axy2-x22-xydx dy I=∫-22∫y-42y+42Axy2-x22-xydx dy I=∫-22y∫y-42y+42A2x-x32-xydxdy I=13184315…

Q: Prove the following result: A function f that is decreasing on [a,b] is integrable on [a,b].

A: Prove the result that, "A function f that is decreasing on[a,b] is integrable on[a,b]''

Question

Calculus Applications of Integration

rates using integration

### Calculus and Integrals: Real-World Applications

### Problem 4: Rate of Fans Entering a Stadium

**Problem Statement:**
Football fans enter a sports stadium at a rate of $ r(t) $ people per hour, where $ t $ is time in hours. If there are 18,000 people in the stadium at the end of the first hour, write an expression that would represent the number of fans inside the stadium by the end of the second hour.

**Solution Approach:**
To find the number of fans inside the stadium by the end of the second hour, integrate the rate function $ r(t) $ over the time interval from 1 hour to 2 hours. Add this to the initial number of fans (18,000) already in the stadium at the end of the first hour.

### Problem 2: Water Treatment Plant Processing Rate

**Problem Statement:**
A water treatment plant processes gray water at a rate given by $ r(t) = 16e^{0.2t} $ million gallons per year, where $ t $ is the time measured in years for $ 0 \leq t \leq 5 $. Write an expression that gives the amount of gray water processed by the treatment plant during the time interval $ 0 \leq t \leq 5 $.

**Solution Approach:**
To determine the total amount of gray water processed over the given time interval, integrate the rate function $ r(t) $ from $ t = 0 $ to $ t = 5 $ years. This will provide the total volume of gray water processed by the plant during this period.

---

#### Detailed Explanation of Diagrams and Graphs:

In both problem statements above, consider plotting the rate functions $ r(t) $ against time to visualize the functional change over the given intervals. For Problem 2 specifically:

- **Graph of $ r(t) = 16e^{0.2t} $**
- **Y-axis:** Rate of gray water processing in million gallons per year.
- **X-axis:** Time $ t $ in years.
- **Curve:** Exponential curve starting from $ r(0) = 16 $ at $ t = 0 $ and increasing as $ t $ grows, reflecting the exponential increase in the rate of water processing.

Such visualizations can aid in understanding how the amount processed

With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Indefinite Integral$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Integrating factors found by inspection. Pls use guide the second picture as solving
INTEGRATION USING PARTIAL FRACTIONS 3x4 + 3x³ - 5x²+x-1 1 x²+x-2 3) dx
4, Suppose SER" is a manifold and pas. Define "To(s) and show that Ipcso= {ucto) | Curve, vrchu)=p_stoff is a parametrized} Tp FI
Optimization Application please refer to image below
integration
Topic: Multiple integrationAnswer the multiple integral I know it is in Spanish, but it is easy to understand. (Including the final result as a check, which is -12)