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You are a process control engineer in a water purification plant. In a certain subprocess, water enters a tank through an inlet at a rate of q; and leaves through an outlet at a rate of q.. Liquid flow is maintained such that q; > q, at any given time. It has been found that the difference between q; and q, is equal to the product of the surface area of the tank bottom A and the rate of change of the liquid column height in the tank h with respect to time. If the variation of h is continuous with respect to time, a. formulate a first-order differential equation to describe the situation. b. It was found that the outlet flow rate depends on the height of the liquid column and is given by q. = 4.429 x 10-3h m³/s. Furthermore, given that A = (0.72) m², q; = 0.003 m³ /s and that h = 1 m when t = 0 s, solve the differential equation formulated (a) to find the relationship between the liquid column height h and the time taken t. c. Verify your answer in (b) using another method of solving the differential equation (e.g. Laplace transforms). d. The rate of change of liquid volume r in the primary tank of the water purification plant was measured for 4 days. If r can be approximated to the function r = 5 sin(0.5t + 0.5) + In (t + 1); where r is in m/day and t is in days, find the approximate volume of water in the tank at 1sts4 the end of the 4 days correct to 3 decimal places using the trapezoidal rule with 4 intervals. e. Calculate the same distance as in (d) above by using the appropriate definite integral. f. Given that the permissible error in the calculation of liquid volume in the primary tank is +0.1 m3, state whether the trapezoidal rule is appropriate for future calculations of the

You are a process control engineer in a water purification plant. In a certain subprocess, water enters a tank through an inlet at a rate of q; and leaves through an outlet at a rate of q.. Liquid flow is maintained such that q; > q, at any given time. It has been found that the difference between q; and q, is equal to the product of the surface area of the tank bottom A and the rate of change of the liquid column height in the tank h with respect to time. If the variation of h is continuous with respect to time, a. formulate a first-order differential equation to describe the situation. b. It was found that the outlet flow rate depends on the height of the liquid column and is given by q. = 4.429 x 10-3h m³/s. Furthermore, given that A = (0.72) m², q; = 0.003 m³ /s and that h = 1 m when t = 0 s, solve the differential equation formulated (a) to find the relationship between the liquid column height h and the time taken t. c. Verify your answer in (b) using another method of solving the differential equation (e.g. Laplace transforms). d. The rate of change of liquid volume r in the primary tank of the water purification plant was measured for 4 days. If r can be approximated to the function r = 5 sin(0.5t + 0.5) + In (t + 1); where r is in m/day and t is in days, find the approximate volume of water in the tank at 1sts4 the end of the 4 days correct to 3 decimal places using the trapezoidal rule with 4 intervals. e. Calculate the same distance as in (d) above by using the appropriate definite integral. f. Given that the permissible error in the calculation of liquid volume in the primary tank is +0.1 m3, state whether the trapezoidal rule is appropriate for future calculations of the

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
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only need part d and part e and part f

4. You are a process control engineer in a water purification plant. In a certain subprocess, water enters a tank through an inlet at a rate of q; and leaves through an outlet at a rate of q. Liquid flow is maintained such that q; > 9, at any given time. It has been found that the difference between q; and q, is equal to the product of the surface area of the tank bottom A and the rate of change of the liquid column height in the tank h with respect to time. If the variation of h is continuous with respect to time, a. formulate a first-order differential equation to describe the situation. b. It was found that the outlet flow rate depends on the height of the liquid column and is (0.72) m², q; = 0.003 m³ /s and that h = 1 m when t = 0 s, solve the differential equation formulated in (a) to find the relationship between the liquid column height h and the time taken t. c. Verify your answer in (b) using another method of solving the differential equation (e.g. given by q, = 4.429 x 10-3h m³/s. Furthermore, given that A = Laplace transforms). d. The rate of change of liquid volume r in the primary tank of the water purification plant was measured for 4 days. If r can be approximated to the function r = 5 sin(0.5t + 0.5) + In (t + 1); 1sts 4 where r is in m?/day and t is in days, find the approximate volume of water in the tank at the end of the 4 days correct to 3 decimal places using the trapezoidal rule with 4 intervals. e. Calculate the same distance as in (d) above by using the appropriate definite integral. f. Given that the permissible error in the calculation of liquid volume in the primary tank is +0.1 m³, state whether the trapezoidal rule is appropriate for future calculations of the same quantity.
Transcribed Image Text:4. You are a process control engineer in a water purification plant. In a certain subprocess, water enters a tank through an inlet at a rate of q; and leaves through an outlet at a rate of q. Liquid flow is maintained such that q; > 9, at any given time. It has been found that the difference between q; and q, is equal to the product of the surface area of the tank bottom A and the rate of change of the liquid column height in the tank h with respect to time. If the variation of h is continuous with respect to time, a. formulate a first-order differential equation to describe the situation. b. It was found that the outlet flow rate depends on the height of the liquid column and is (0.72) m², q; = 0.003 m³ /s and that h = 1 m when t = 0 s, solve the differential equation formulated in (a) to find the relationship between the liquid column height h and the time taken t. c. Verify your answer in (b) using another method of solving the differential equation (e.g. given by q, = 4.429 x 10-3h m³/s. Furthermore, given that A = Laplace transforms). d. The rate of change of liquid volume r in the primary tank of the water purification plant was measured for 4 days. If r can be approximated to the function r = 5 sin(0.5t + 0.5) + In (t + 1); 1sts 4 where r is in m?/day and t is in days, find the approximate volume of water in the tank at the end of the 4 days correct to 3 decimal places using the trapezoidal rule with 4 intervals. e. Calculate the same distance as in (d) above by using the appropriate definite integral. f. Given that the permissible error in the calculation of liquid volume in the primary tank is +0.1 m³, state whether the trapezoidal rule is appropriate for future calculations of the same quantity.
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