Explanation Given: The object is of mass 160 lbs was dropped 500 feet down into a 75 foot tall tank of liquid. Formula used: The differential equation for calculating the time and impact velocity is given below. v = m k g − m k g e − k m t where k is coefficient of air resistance. Calculation: When an object falls through the air its velocity can be formulated by the differential equation: m d v d t = m g − k v The above equation can be also written as, d v d t + k m v = g … … ( 1 ) Solve the above differential equation by first order differential equation. Here, Integrating factor = e ∫ k m d t = e k m t Multiply the equation (1) with the integrating factor and integrate. Therefore, e k m t d v d t + e k m t k m v = g e k m t d d t ( v e k m t ) = g e k m t v = m k g + c e − k m t … … ( 2 ) The velocity of the object, as it leaves the helicopter is zero. So, use the initial condition v ( 0 ) = 0 to determine the constant c . v = m k g + c e − k m t 0 = m k g + c e − k m .0 c = − m k g Substitute c = − m k g for c in the equation (2). Then, the equation ( 2 ) becomes, v = m k g − m k g e − k m t . Substitute 160 lbs for m , 32 ft/s 2 for g and 0.25 for k in the above equation to calculate the velocity of the object. v = 20480 − 20480 e − 1 640 t d s d t = 20480 − 20480 e − 1 640 t Integrate the above equation to find the position equation. ∫ d s = ∫ ( 20480 − 20480 e − 1 640 t ) d t s = 20480 t + 13107200 e − 1 640 t + d The position of the object at the level of the helicopter is zero. Therefore, use the initial condition s ( 0 ) = 0 in the above equation to get the constant term d s = 20 , 480 t + 13 , 107 , 200 e − 1 640 t + d 0 = 20 , 480 × 0 + 13 , 107 , 200 e − 1 640 .0 + d d = − 13 , 107 , 200 Therefore the position equation of the object is given by, s = 20 , 480 t + 13 , 107 , 200 e − 1 640 t − 13 , 107 , 200 The object enters the tank of liquid when it covered a distance of 500 feet . Then, substitute 500 feet for s in position equation, s = 20480 t + 13107200 e − 1 640 t − 13107200 500 = 20480 t + 13107200 e − 1 640 t − 13107200 Solve the above equation for t

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 2.8, Problem 30E

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ם Hwk 25 Hwk 25 - (MA 244-03) (SP25) || X Answered: [) Hwk 25 Hwk 28 - (X + https://www.webassign.net/web/Student/Assignment-Responses/last?dep=36606604 3. [1.14/4 Points] DETAILS MY NOTES LARLINALG8 6.4.013. Let B = {(1, 3), (-2, -2)} and B' = {(−12, 0), (-4, 4)} be bases for R², and let 42 - [13] A = 30 be the matrix for T: R² R² relative to B. (a) Find the transition matrix P from B' to B. 6 4 P = 9 4 (b) Use the matrices P and A to find [v] B and [T(V)] B, where [v]B[31]. 26 [V] B = -> 65 234 [T(V)]B= -> 274 (c) Find P-1 and A' (the matrix for T relative to B'). -1/3 1/3 - p-1 = -> 3/4 -1/2 ↓ ↑ -1 -1.3 A' = 12 8 ↓ ↑ (d) Find [T(v)] B' two ways. 4.33 [T(v)]BP-1[T(v)]B = 52 4.33 [T(v)]B' A'[V]B' = 52 目 67% PREVIOUS ANSWERS ill ASK YOUR TEACHER PRACTICE ANOTHER

[) Hwk 25 Hwk 28 - (MA 244-03) (SP25) || X Success Confirmation of Questic X + https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=36606607&tags=autosave#question 384855 DETAILS MY NOTES LARLINALG8 7.2.001. 1. [-/2.85 Points] Consider the following. -14 60 A = [ -4-5 P = -3 13 -1 -1 (a) Verify that A is diagonalizable by computing P-1AP. P-1AP = 具首 (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n x n matrices, then they have the same eigenvalues. (11, 12) = Need Help? Read It SUBMIT ANSWER 2. [-/2.85 Points] DETAILS MY NOTES LARLINALG8 7.2.007. For the matrix A, find (if possible) a nonsingular matrix P such that P-1AP is diagonal. (If not possible, enter IMPOSSIBLE.) P = A = 12 -3 -4 1 Verify that P-1AP is a diagonal matrix with the eigenvalues on the main diagonal. P-1AP = Need Help? Read It Watch It SUBMIT ANSWED 80% ill จ ASK YOUR TEACHER PRACTICE ANOTHER ASK YOUR…

Chapter 2 Solutions

Advanced Engineering Mathematics

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The time and the impact velocity when the object hits the bottom of the tank.

Solution Summary: The author explains the differential equation for calculating the time and impact velocity when the object hits the bottom of the tank.

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Chapter 2 Solutions