(minutes) 0 4 9 W(t) (degrees Fahrenheit) 55.0 57.1 61.8 15 20 67.9 71.0 The temperature of water in a tub at time is modeled by a strictly increasing, twice-differentiable function W, where W (1) is measured in degrees Fahrenheit and r is measured in minutes. At time f = 0, the temperature of the water is 55°F. The water is heated for 30 minutes, beginning at time=0. Values of W(1) at selected timest for the first 20 minutes are given in the table above. (a) Use the data in the table to estimate W (12). Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem. (b) Use the data in the table to evaluate [W(1) dt. Using correct units, interpret the meaning of '(1) dt in the context of this problem. 1 (c) For 0 ≤ I ≤ 20, the average temperature of the water in the tub is W(1) dr. Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate W(t) dr. Does this approximation overestimate or underestimate the average temperature of the water over these 20 minutes? Explain your reasoning. (d) For 20 St≤ 25, the function W that models the water temperature has first derivative given by W'(t) = 0.4√f cos (0.06r). Based on the model, what is the temperature of the water at time = 25?
(minutes) 0 4 9 W(t) (degrees Fahrenheit) 55.0 57.1 61.8 15 20 67.9 71.0 The temperature of water in a tub at time is modeled by a strictly increasing, twice-differentiable function W, where W (1) is measured in degrees Fahrenheit and r is measured in minutes. At time f = 0, the temperature of the water is 55°F. The water is heated for 30 minutes, beginning at time=0. Values of W(1) at selected timest for the first 20 minutes are given in the table above. (a) Use the data in the table to estimate W (12). Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem. (b) Use the data in the table to evaluate [W(1) dt. Using correct units, interpret the meaning of '(1) dt in the context of this problem. 1 (c) For 0 ≤ I ≤ 20, the average temperature of the water in the tub is W(1) dr. Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate W(t) dr. Does this approximation overestimate or underestimate the average temperature of the water over these 20 minutes? Explain your reasoning. (d) For 20 St≤ 25, the function W that models the water temperature has first derivative given by W'(t) = 0.4√f cos (0.06r). Based on the model, what is the temperature of the water at time = 25?
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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Transcribed Image Text:t (minutes)
0
W(t) (degrees Fahrenheit) 55.0
4
9
57.1 61.8
15
20
67.9 71.0
The temperature of water in a tub at time t is modeled by a strictly increasing, twice-differentiable function W,
where W (1) is measured in degrees Fahrenheit and is measured in minutes. At time = 0, the temperature of
the water is 55°F. The water is heated for 30 minutes, beginning at time t = 0. Values of W(1) at selected
times for the first 20 minutes are given in the table above.
(a) Use the data in the table to estimate W (12). Show the computations that lead to your answer. Using correct
units, interpret the meaning of your answer in the context of this problem.
(b) Use the data in the table to evaluate [20 W (1) dr. Using correct units, interpret the meaning of [20W (1) dt
in the context of this problem.
(c) For 0 ≤ t ≤ 20, the average temperature of the water in the tub is 2 W(1) dt. Use a left Riemann sum
with the four subintervals indicated by the data in the table to approximate
W(1) dt. Does this
approximation overestimate or underestimate the average temperature of the water over these 20 minutes?
Explain your reasoning.
(d) For 20 St ≤ 25, the function W that models the water temperature has first derivative given by
W'(t) = 0.4√t cos (0.06r). Based on the model, what is the temperature of the water at time / = 25 ?
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