Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
As a raindrop falls, it evaporates while retaining its spherical shape. If we make the further assumptions that the rate at which the raindrop evaporates is proportional to its surface area and that air resistance is negligible, then a model for the velocity v(t) of the raindrop is 3(k/p) (k/p)t + ro Here p is the density of water, ro is the radius of the raindrop at t= 0, k < 0 is the constant of proportionality, and the downward direction is taken to be the positive direction. (a) Solve for v(t) if the raindrop falls from rest. v(t) = dv dt -v=g. (b) This model assumes that the rate at which the raindrop evaporates-that is, the rate at which it loses mass-is proportional to its surface area, with constant of proportionality k<0. This assumption implies that the rate at which the radius r of the raindrop decreases is a constant. Show that the radius of the raindrop at time t is r(t) = (k/p)t + 50₁ Letting A denote the surface area of the raindrop and m the mass, the model assumes that and r. dmk A. dt Since mass equals density times Select and a sphere of radius r has volume m = A- and surface area we obtain the following formulas form and A in terms of p

As a raindrop falls, it evaporates while retaining its spherical shape. If we make the further assumptions that the rate at which the raindrop evaporates is proportional to its surface area and that air resistance is negligible, then a model for the velocity v(t) of the raindrop is 3(k/p) (k/p)t + ro Here p is the density of water, ro is the radius of the raindrop at t= 0, k < 0 is the constant of proportionality, and the downward direction is taken to be the positive direction. (a) Solve for v(t) if the raindrop falls from rest. v(t) = dv dt -v=g. (b) This model assumes that the rate at which the raindrop evaporates-that is, the rate at which it loses mass-is proportional to its surface area, with constant of proportionality k<0. This assumption implies that the rate at which the radius r of the raindrop decreases is a constant. Show that the radius of the raindrop at time t is r(t) = (k/p)t + 50₁ Letting A denote the surface area of the raindrop and m the mass, the model assumes that and r. dmk A. dt Since mass equals density times Select and a sphere of radius r has volume m = A- and surface area we obtain the following formulas form and A in terms of p

Advanced Engineering Mathematics
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
icon
Related questions
Question
100%

Hi bartleby expert! Please help me asap,  I will give an upvote.  

This is given to us but I'm having hard time guessing what's supposed to be in the blank. Hope you can help :)

Thank you! 

 

As a raindrop falls, it evaporates while retaining its spherical shape. If we make the further assumptions that the rate at which the raindrop evaporates is proportional to its surface area and that air resistance is negligible, then a model for the velocity v(t) of the raindrop is 3(k/p) -v = g. dv dt (k/p)t + ro Here p is the density of water, ro is the radius of the raindrop at t = 0, k < 0 is the constant of proportionality, and the downward direction is taken to be the positive direction. (a) Solve for v(t) if the raindrop falls from rest. v(t) = (b) This model assumes that the rate at which the raindrop evaporates-that is, the rate at which it loses mass-is proportional to its surface area, with constant of proportionality k < 0. This assumption. implies that the rate at which the radius r of the raindrop decreases is a constant. Show that the radius of the raindrop at time t is r(t) = (k/p)t + ro. Letting A denote the surface area of the raindrop and m the mass, the model assumes that dm dt and r. Since mass equals density times ---Select-- and a sphere of radius r has volume = k. A. m = A = and surface area , we obtain the following formulas for m and A in terms of p
Transcribed Image Text:As a raindrop falls, it evaporates while retaining its spherical shape. If we make the further assumptions that the rate at which the raindrop evaporates is proportional to its surface area and that air resistance is negligible, then a model for the velocity v(t) of the raindrop is 3(k/p) -v = g. dv dt (k/p)t + ro Here p is the density of water, ro is the radius of the raindrop at t = 0, k < 0 is the constant of proportionality, and the downward direction is taken to be the positive direction. (a) Solve for v(t) if the raindrop falls from rest. v(t) = (b) This model assumes that the rate at which the raindrop evaporates-that is, the rate at which it loses mass-is proportional to its surface area, with constant of proportionality k < 0. This assumption. implies that the rate at which the radius r of the raindrop decreases is a constant. Show that the radius of the raindrop at time t is r(t) = (k/p)t + ro. Letting A denote the surface area of the raindrop and m the mass, the model assumes that dm dt and r. Since mass equals density times ---Select-- and a sphere of radius r has volume = k. A. m = A = and surface area , we obtain the following formulas for m and A in terms of p
Plugging these formulas into the differential equation gives: dt ). . 55 55 55 r(t) = Plugging in r(0) = ro gives C = dr dt dr dt dr = k· 4πr² = k· 4πr² = k = dt Integrating the above differential equation gives r(t) = Je + C. (assuming r = 0) , so that (c) If ro = 0.05 ft and r = 0.008 ft 10 seconds after the raindrop falls from a cloud, determine the time at which the raindrop has evaporated completely. (Round your answer to one decimal place.) sec
Transcribed Image Text:Plugging these formulas into the differential equation gives: dt ). . 55 55 55 r(t) = Plugging in r(0) = ro gives C = dr dt dr dt dr = k· 4πr² = k· 4πr² = k = dt Integrating the above differential equation gives r(t) = Je + C. (assuming r = 0) , so that (c) If ro = 0.05 ft and r = 0.008 ft 10 seconds after the raindrop falls from a cloud, determine the time at which the raindrop has evaporated completely. (Round your answer to one decimal place.) sec
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question

Hello again!  Can you double check the answer of the previous bartleby expert?  I'm confused at some steps. I will upvote this again. Thank you. 

Solution
Bartleby Expert
SEE SOLUTION
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS