Explanation Given information: A sample of bismuth Bismuth − 210 that initially had a mass of 1000 mg decayed 500 mg in 5 days . Formula used: d N d t = N Calculation: Consider the general differential equation. − d N d t = k N d N N = k d t Now apply set the initial conditions as shown. N ( 0 ) = 1000 mg N ( 5 ) = 500 mg Integrate both sides with the initial conditions. ∫ 1000 500 d N N = − k ∫ 0 5 d t | ln N | 1000 500 = − 5 k ln 500 − ln 1000 = − 5 k ln 500 1000 = − 5 k −

4th Edition

ISBN: 9781319279813

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 9.1, Problem 54E

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Set up and solve an initial value problem for determine by M(t).

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Chapter 9.1, Problem 53E

Chapter 9.1, Problem 55E

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(12 points) Let E={(x, y, z)|x²+ y² + z² ≤ 4, x, y, z > 0}. (a) (4 points) Describe the region E using spherical coordinates, that is, find p, 0, and such that (x, y, z) (psin cos 0, psin sin 0, p cos) € E. (b) (8 points) Calculate the integral E xyz dV using spherical coordinates.

(10 points) Let f(x, y, z) = ze²²+y². Let E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z < 3}. Calculate the integral y, f(x, y, z) dV.

(14 points) Let f: R3 R and T: R3. →R³ be defined by f(x, y, z) = ln(x²+ y²+2²), T(p, 0,4)=(psin cos 0, psin sin, pcos). (a) (4 points) Write out the composition g(p, 0, 4) = (foT)(p,, ) explicitly. Then calculate the gradient Vg directly, i.e. without using the chain rule. (b) (4 points) Calculate the gradient Vf(x, y, z) where (x, y, z) = T(p, 0,4). (c) (6 points) Calculate the derivative matrix DT(p, 0, p). Then use the Chain Rule to calculate Vg(r,0,4).

Chapter 9 Solutions

CALCULUS:EARLY TRANS.-SAPLINGPLUS

Ch. 9.1 - Prob. 1PQ Ch. 9.1 - Prob. 2PQ Ch. 9.1 - Prob. 3PQ Ch. 9.1 - Prob. 4PQ Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E

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Chapter 9 Solutions