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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 dr (k/p)t + rog. 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 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-k.A. Since mass equals density times ---Select--and a sphere of radius r has volume m- A- Plugging these formulas into the differential equation gives: ( - k-4m² ).- k-4m² Plugging in r(0) rogives C- -k (assuming r = 0) Integrating the above differential equation gives so that and surface area we obtain the following formulas form and A in terms of p and r. (c) If ro - 0.04 ft and r= 0.006 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

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 dr (k/p)t + rog. 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 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-k.A. Since mass equals density times ---Select--and a sphere of radius r has volume m- A- Plugging these formulas into the differential equation gives: ( - k-4m² ).- k-4m² Plugging in r(0) rogives C- -k (assuming r = 0) Integrating the above differential equation gives so that and surface area we obtain the following formulas form and A in terms of p and r. (c) If ro - 0.04 ft and r= 0.006 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

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
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12. This subject is a differential equation. Please answer need 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) + (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) = = 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 + ro. Letting A denote the surface area of the raindrop and m the mass, the model assumes that dmk A. dt Since mass equals density times ---Select--- and a sphere of radius r has volume m = Plugging these formulas into the differential equation gives: ( 99 99 99 Plugging in r(0) = ro gives C = r(t) = =k. 47² = k· 4m² Integrating the above differential equation gives r(t) = = k (assuming r = 0) t + C. so that and surface area. , we obtain the following formulas for m and A in terms of p and r. (c) If ro= 0.04 ft and r = 0.006 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: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) = = 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 + ro. Letting A denote the surface area of the raindrop and m the mass, the model assumes that dmk A. dt Since mass equals density times ---Select--- and a sphere of radius r has volume m = Plugging these formulas into the differential equation gives: ( 99 99 99 Plugging in r(0) = ro gives C = r(t) = =k. 47² = k· 4m² Integrating the above differential equation gives r(t) = = k (assuming r = 0) t + C. so that and surface area. , we obtain the following formulas for m and A in terms of p and r. (c) If ro= 0.04 ft and r = 0.006 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
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