To determine The maximum amount of fertilizer in the tank and also the time required to reach the maximum. Explanation Formula used: The differential equation describing the mixing process is given by the formula d y d t = ( chemical's arrival rate ) − y ( t ) V ( t ) ⋅ ( outflow rate ) . Calculation: Let y ( t ) be the amount of fertilizer in the tank at time t . The initial amount of water in the tank is 100 gal. So, the amount of fertilizer is 0 gal. Thus, the initial condition is y ( 0 ) = 0 . The solution containing 1 lb/gal of fertilizer enters into the tank at the rate of 1 gal/min and pumped out at a rate of 3 gal/min. Evaluate the arrival rate of the arrival of fertilizer as follows. Arrival rate = ( 1 lb gal ) ( 1 gal min ) = 1 ( lb min ) Thus, the arrival rate of the concentrate is 1 ( lb min ) . Evaluate the rate of change of the amount of fertilizer in tank is evaluated as follows. ( Rate of change of amount in container ) = ( rate at which chemical arrives ) − ( rate at which chemical departs ) = 1 gal − 3 gal = − 2 gal Thus, 2 gal of fertilizer is pumped out from the tank for every 1 min. At time t the volume of the tank is given by ( 100 − 2 t ) gal . Evaluate the differential equation describing the mixing process as follows. d y d t = ( chemical's arrival rate ) − y ( t ) V ( t ) ⋅ ( outflow rate ) d y d t = 1 − y 100 − 2 t ( 3 ) d y d t + 3 100 − 2 t y = 1 Thus, the initial value problem is d y d t + 3 100 − 2 t y = 1 , y ( 0 ) = 0 . Compare the above equation with the standard equation d y d t + P ( x ) y = Q ( x ) . Here, P ( t ) = 3 100 − 2 t and Q ( t ) = 1 . Evaluate the integrating factor as follows. v ( t ) = e ∫ P ( t ) d t = e ∫ 3 100 − 2 t d t = e − 3 2 ln ( 100 − 2 t ) = e ln ( 100 − 2 t ) − 3 2 = ( 100 − 2 t ) − 3 2 Evaluate the solution as follows. y = 1 v ( t ) ∫ v ( t ) Q ( t ) d t = 1 ( 100 − 2 t ) − 3 2 ∫ ( 100 − 2 t ) − 3 2 ( 1 ) d t = 1 ( 100 − 2 t ) − 3 2 ( − 2 ( 100 − 2 t ) − 1 2 − 2 + C ) = ( 100 − 2 t ) 3 2 ( 100 − 2 t ) − 1 2 + C ( 100 − 2 t ) 3 2 = ( 100 − 2 t ) + C ( 100 − 2 t ) 3 2 The initial amount of fertilizer in the tank is 0 gal

14th Edition

ISBN: 9781323835029

Author: WEIR

Publisher: PEARSON C

See similar textbooks

Concept explainers

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 …

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experime…

Videos

Question

Chapter 9.3, Problem 15E

To determine

The maximum amount of fertilizer in the tank and also the time required to reach the maximum.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.3, Problem 14E

Chapter 9.3, Problem 16E

Students have asked these similar questions

4 3 2 -5 4-3 -2 -1 1 2 3 4 5 12 23 -4 The function graphed above is: Increasing on the interval(s) Decreasing on the interval(s)

Question 4 The plot below represents the function f(x) 8 7 3 pts O -4-3-2-1 6 5 4 3 2 + 1 2 3 5 -2+ Evaluate f(3) f(3) = Solve f(x) = 3 x= Question 5

Question 14 6+ 5 4 3 2 -8-2 2 3 4 5 6 + 2 3 4 -5 -6 The graph above is a transformation of the function f(x) = |x| Write an equation for the function graphed above g(x) =

Chapter 9 Solutions

MYMATHLAB ACCESS FOR CALCULUS >I< 2018

Ch. 9.1 - In Exercises 1–4, match the differential equations...Ch. 9.1 - In Exercises 1–4, match the differential equations...Ch. 9.1 - In Exercises 1–4, match the differential equations...Ch. 9.1 - In Exercises 1–4, match the differential equations...Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E Ch. 9.1 - Prob. 7E Ch. 9.1 - Prob. 8E Ch. 9.1 - In Exercises 7–12, write an equivalent first-order...Ch. 9.1 - Prob. 10E

Knowledge Booster

$Calculus$

Learn more about

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