The function [tex]$W(t)=40 e ^{-0.35 t}$[/tex] models the healing of a wound after [tex]$t$[/tex] hours, where [tex]$W(t)$[/tex] represents the wound in square millimeters. What relationship do [tex]$W(t)$[/tex] and [tex]$t$[/tex] have?
A. when [tex]$t=0, W(t)\ \textgreater \ 40$[/tex]
B. when [tex]$t=0, W(t)\ \textless \ 40$[/tex]
C. when [tex]$t\ \textgreater \ 0, W(t)\ \textgreater \ 40$[/tex]
D. when [tex]$t\ \textgreater \ 0, W(t)\ \textless \ 40$[/tex]
Answer (2)
when 0, W(t)<40"> t > 0 , W ( t ) < 40 . 0, \ W(t)<40}"> w h e n t > 0 , W ( t ) < 40
Explanation
Understanding the Problem We are given the function W ( t ) = 40 e − 0.35 t which models the healing of a wound after t hours. We need to determine the relationship between W ( t ) and t , specifically when t = 0 and when 0"> t > 0 , and compare W ( t ) to 40.
Calculating W(0) First, let's find the value of W ( t ) when t = 0 . We substitute t = 0 into the function: W ( 0 ) = 40 e − 0.35 ( 0 ) = 40 e 0
Since e 0 = 1 , we have: W ( 0 ) = 40 ( 1 ) = 40 So, when t = 0 , W ( t ) = 40 . Therefore, the statement 'when 40"> t = 0 , W ( t ) > 40 ' is false, and the statement 'when t = 0 , W ( t ) < 40 ' is also false.
Analyzing W(t) for t>0 Now, let's analyze the behavior of W ( t ) when 0"> t > 0 . The function W ( t ) = 40 e − 0.35 t has an exponential term e − 0.35 t . Since − 0.35 is negative, the exponent is negative for 0"> t > 0 . As t increases, − 0.35 t becomes more negative, and e − 0.35 t approaches 0. Also, for 0"> t > 0 , e − 0.35 t will be less than e 0 = 1 . Therefore, for 0"> t > 0 , e − 0.35 t < 1 .
Comparing W(t) with 40 for t>0 Since e − 0.35 t < 1 for 0"> t > 0 , we can multiply both sides of the inequality by 40: 40 e − 0.35 t < 40 ( 1 ) W ( t ) < 40 So, when 0"> t > 0 , W ( t ) < 40 . Therefore, the statement 'when 0, W(t) > 40"> t > 0 , W ( t ) > 40 ' is false, and the statement 'when 0, W(t) < 40"> t > 0 , W ( t ) < 40 ' is true.
Final Answer Based on our analysis, the correct statement is: when 0, W(t) < 40"> t > 0 , W ( t ) < 40 .
Examples
This model can be used in healthcare to estimate the size of a wound over time and predict healing rates. For example, if a doctor measures a wound to be 40 square millimeters initially, they can use this model to predict that after a certain number of hours, the wound will be smaller than 40 square millimeters. This helps in planning treatment and monitoring patient recovery.
The function W ( t ) = 40 e − 0.35 t shows that at t = 0 , W ( t ) = 40 . For any positive t , W ( t ) is always less than 40, indicating the wound heals over time. Therefore, the correct answer is option D: when 0, W(t) < 40"> t > 0 , W ( t ) < 40 .
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