17 RE 1.1.10 Problem 1017. A pond initially contains 1,000,000 gal of water and an unknownamount of an undesirable chemical. Water containing 0.01 grams ofthis chemical per gallon flows into the pond at a rate of 300 gal/h. Themixture flows out at the same rate, so the amount of water in the pondremains constant. Assume that the chemical is uniformly distributedthroughout the pond.a. Write a differential equation for the amount of chemical inthe pond at any time.b. How much of the chemical will be in the pond after a verylong time? Does this limiting amount depend on the amount thatwas present initially?c. Write a differential equation for the concentration of thechemical in the pond at time t. Hint: The concentration isc = a/v = a(t)/106.18. A spherical raindrop evaporates at a rate proportional to itssurface area. Write a differential equation for the volume of theraindrop as a function of time.19. Newton's law of cooling states that the temperature of anobject changes at a rate proportional to the difference between thetemperature of the object itself and the temperature of its surroundings(the ambient air temperature in most cases). Suppose that the ambienttemperature is 70°F and that the rate constant is 0.05 (min)-¹. Write adifferential equation for the temperature of the object at any time. Notethat the differential equation is the same whether the temperature ofthe object is above or below the ambient temperature.1.2ForthatIn the preceding section we derived the differential equationsIn egivebehof yof thcanGGGGD2 SeeAeroMoniSolutions of Some Differentia

dQ /dt=(rate in) - (rate out) The Rate of Change, is defined as Q(t)=[(flow rate of the liquid…

17. A pond initially contains 1,000,000 gal of water and an unknown amount of an undesirable chemical. Water containing 0.01 grams of this chemical per gallon flows into the pond at a rate of 300 gal/h. The mixture flows out at the same rate, so the amount of water in the pond remains constant. Assume that the chemical is uniformly distributed throughout the pond. a. Write a differential equation for the amount of chemical in the pond at any time. b. How much of the chemical will be in the pond after a very long time? Does this limiting amount depend on the amount that was present initially? c. Write a differential equation for the concentration of the chemical in the pond at time t. Hint: The concentration is c = a/v = a(t)/ 106.

17. A pond initially contains 1,000,000 gal of water and an unknown amount of an undesirable chemical. Water containing 0.01 grams of this chemical per gallon flows into the pond at a rate of 300 gal/h. The mixture flows out at the same rate, so the amount of water in the pond remains constant. Assume that the chemical is uniformly distributed throughout the pond. a. Write a differential equation for the amount of chemical in the pond at any time. b. How much of the chemical will be in the pond after a very long time? Does this limiting amount depend on the amount that was present initially? c. Write a differential equation for the concentration of the chemical in the pond at time t. Hint: The concentration is c = a/v = a(t)/ 106.

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter11: Nonlinear Programming

Section11.2: Introductory Concepts

Problem 3P

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Concept explainers

Rod Cutting

A naïve solution for the above-stated problem is to evaluate all of the possible combinations of different pieces of varying size to find the maximum revenue-generating combination. Although this method seems the easiest and quick to practice, it has exponential time complexity. So there is a need to figure out a better method than this.

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RE 1.1.10 Problem 10
17. A pond initially contains 1,000,000 gal of water and an unknown
amount of an undesirable chemical. Water containing 0.01 grams of
this chemical per gallon flows into the pond at a rate of 300 gal/h. The
mixture flows out at the same rate, so the amount of water in the pond
remains constant. Assume that the chemical is uniformly distributed
throughout the pond.
a. Write a differential equation for the amount of chemical in
the pond at any time.
b. How much of the chemical will be in the pond after a very
long time? Does this limiting amount depend on the amount that
was present initially?
c. Write a differential equation for the concentration of the
chemical in the pond at time t. Hint: The concentration is
c = a/v = a(t)/106.
18. A spherical raindrop evaporates at a rate proportional to its
surface area. Write a differential equation for the volume of the
raindrop as a function of time.
19. Newton's law of cooling states that the temperature of an
object changes at a rate proportional to the difference between the
temperature of the object itself and the temperature of its surroundings
(the ambient air temperature in most cases). Suppose that the ambient
temperature is 70°F and that the rate constant is 0.05 (min)-¹. Write a
differential equation for the temperature of the object at any time. Note
that the differential equation is the same whether the temperature of
the object is above or below the ambient temperature.
1.2
For
that
In the preceding section we derived the differential equations
In e
give
beh
of y
of th
can
G
G
G
G
D
2 See
Aero
Moni
Solutions of Some Differentia

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