A tank in the shape of a cylinder standing on end leaks water through a circular hole in itsbottom. If friction is considered, the height h of the water in the tank as a function of timeis given bydhAn 2ghAwwhere Au and An are the cross-sectional areas of the water in the tank and of the hole,respectively, g is the gravitational coefficient, and 0

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Answered: bottom. If friction is considered, the…

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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The density of air changes with height. Under some conditions density p, depends on height z, and temperature T according to the following equation where Po and A are both constants. A meteorological balloon ascends (i.e., starts at z = 1 and gains height) over the course of several hours. Complete parts (a) and (b) below. Az P(z,T) = Po e ..... dz v) and that the temperature changes over time (i.e., that T is given by a function T(t)), derive, using the chain rule, an expression for the rate of change of air density, (a) Assuming that the balloon ascends at a speed v (i.e., dt as measured by the weather balloon. Choose the correct answer below. dp Az dT O A. dt T2 dt dp %3D dt Az dT dp С. dt %3D + T dt dp Az) dT O D. dt T2) dt (b) Assume that v = 1, Po = 1, and A = 1 and that when t= 0, T= 1. Are there any conditions under which the density, as measured by the balloon will not change in time? That is, find a differential equation that T must satisfy, if dp = 0, and solve this…
After a diver jumps off a diving board, the edge of the board oscillates with position given by s(t) = −5cos(t) cm at t seconds after the jump. Part I: Complete the following steps: Using the GeoGebra tool in Canvas, sketch one period of the position function for t ≥ 0. Find the velocity function. Using the GeoGebra tool in Canvas, sketch one period of the velocity function for t ≥ 0. Determine the times in the first period (or a single period) when the velocity is 0. Find the acceleration function. Using the GeoGebra tool in Canvas, sketch one period of the acceleration function for t ≥ 0. Save your GeoGebra work as a .pdf file for submission. Part II: Based on your work in Part I, discuss the following: Discuss any observations you can make about the relationships between the graphs of the position, velocity, and acceleration functions. In particular, make sure to include the following: The connections between critical points on each graph. The similarities and differences among…
(e) Which function has a second derivative that must be always positive? How do you know?
Automobile traffic passes a point P on a road of width w feet with an average rate of R vehicles per second. Although the arrival of automobiles is irregular, traffic engineers have found that the average waiting time T until there is a gap in traffic of at least 1 seconds is approximately T = te Riseconds. A pedestrian walking at a speed of 3.3 ft/s requires t =s to cross the road. Therefore, the average time the pedestrian will have to wait before crossing is f(w, R) = () WR/3.3 What is the pedestrian's average waiting time if w = 24 ft and R = 0.2 vehicle per second? (Use decimal notation. Give your answer to two decimal places.) (= Use the Linear Approximation to estimate the increase in waiting time if w is increased to 26 ft. (Use decimal notation. Give your answer to two decimal places.) Af = 31.15 t = Estimate the waiting time if the width is increased to 26 ft and R decreases to 0.18. (Use decimal notation. Give your answer to two decimal places.) 6.37 A = 32.48 Incorrect What…
1. The accompanying figure shows the graph of the velocity (ft / sec) of a model rocket for the first 8 sec after launch. The rocket ac- celerated straight up for the first 2 sec and then coasted to reach its maximum height at t = 8 sec. 200 150 100 50 2 4 6 Time after launch (sec) a. Assuming that the rocket was launched from ground level, about how high did it go? (This is the rocket in Section 3.3, Exercise 17, but you do not need to do Exercise 17 to do the exercise here.) b. Sketch a graph of the rocket's height above ground as a func- tion of time for 0 < t< 8. Velocity (ft/sec)

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