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The pressure of the oil in a reservoir tends to drop with time. By taking sample pressure readings for a particular oil reservoir, petroleum engineers have found that the change in the function P(t)=1³-3212 +256t, where t is time in years from the date of the first reading. (a) Find the following: P(0), P(4), P(15), P(17) (b) Graph P(t) (c) Over what period is the change (drop) in pressure increasing? decreasing? (a) Evaluate P(0), P(4), P(15), and P(17) P(0)= C...

The pressure of the oil in a reservoir tends to drop with time. By taking sample pressure readings for a particular oil reservoir, petroleum engineers have found that the change in the function P(t)=1³-3212 +256t, where t is time in years from the date of the first reading. (a) Find the following: P(0), P(4), P(15), P(17) (b) Graph P(t) (c) Over what period is the change (drop) in pressure increasing? decreasing? (a) Evaluate P(0), P(4), P(15), and P(17) P(0)= C...

Advanced Engineering Mathematics
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 pressure of the oil in a reservoir tends to drop with time. By taking sample pressure readings for a particular oil reservoir, petroleum engineers have found that the change in pressure is given by the function \( P(t) = t^3 - 32t^2 + 256t \), where \( t \) is time in years from the date of the first reading. (a) Find the following: \( P(0) \), \( P(4) \), \( P(15) \), \( P(17) \). (b) Graph \( P(t) \). (c) Over what period is the change (drop) in pressure increasing? decreasing? 1. **(a) Evaluate \( P(0) \), \( P(4) \), \( P(15) \), and \( P(17) \):** - \( P(0) = \) ______ To tackle this problem, calculate the values of the pressure function at the specified times through substitution. 2. **(b) Graph \( P(t) \):** - Plot the function \( P(t) = t^3 - 32t^2 + 256t \) over an appropriate domain to visualize the changes in pressure over time. 3. **(c) Analyze the Period of Pressure Change:** - Determine the intervals of \( t \) over which the pressure decreases or increases. Consider the behavior of the derivative \( P'(t) \) to find these intervals, as changes in the sign of the derivative indicate changes in monotonicity. For a more thorough understanding, sketching the graph of \( P(t) \) and evaluating the critical points will be useful in determining the nature of each interval.
Transcribed Image Text:The pressure of the oil in a reservoir tends to drop with time. By taking sample pressure readings for a particular oil reservoir, petroleum engineers have found that the change in pressure is given by the function \( P(t) = t^3 - 32t^2 + 256t \), where \( t \) is time in years from the date of the first reading. (a) Find the following: \( P(0) \), \( P(4) \), \( P(15) \), \( P(17) \). (b) Graph \( P(t) \). (c) Over what period is the change (drop) in pressure increasing? decreasing? 1. **(a) Evaluate \( P(0) \), \( P(4) \), \( P(15) \), and \( P(17) \):** - \( P(0) = \) ______ To tackle this problem, calculate the values of the pressure function at the specified times through substitution. 2. **(b) Graph \( P(t) \):** - Plot the function \( P(t) = t^3 - 32t^2 + 256t \) over an appropriate domain to visualize the changes in pressure over time. 3. **(c) Analyze the Period of Pressure Change:** - Determine the intervals of \( t \) over which the pressure decreases or increases. Consider the behavior of the derivative \( P'(t) \) to find these intervals, as changes in the sign of the derivative indicate changes in monotonicity. For a more thorough understanding, sketching the graph of \( P(t) \) and evaluating the critical points will be useful in determining the nature of each interval.
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