Solve for t.
[tex]e^{-0.65 t}=0.61[/tex]
Answer (2)
Take the natural logarithm of both sides: ln ( e − 0.65 t ) = ln ( 0.61 ) .
Simplify using logarithm properties: − 0.65 t = ln ( 0.61 ) .
Isolate t : t = − 0.65 l n ( 0.61 ) .
Calculate and round: t ≈ 0.7605 .
Explanation
Problem Analysis We are given the equation e − 0.65 t = 0.61 and asked to solve for t .
Solving for t To solve for t , we first take the natural logarithm of both sides of the equation: ln ( e − 0.65 t ) = ln ( 0.61 ) Using the property of logarithms, we simplify the left side: − 0.65 t = ln ( 0.61 ) Now, we divide both sides by − 0.65 to isolate t :
t = − 0.65 ln ( 0.61 )
Calculating the Value of t Using a calculator, we find that t = − 0.65 ln ( 0.61 ) ≈ − 0.65 − 0.494296 ≈ 0.7604558797 Rounding to four decimal places, we get t ≈ 0.7605 .
Final Answer Therefore, the solution for t rounded to four decimal places is 0.7605 .
Examples
Exponential decay problems like this are used in various fields. For example, in radioactive decay, the amount of a substance decreases exponentially over time. If you know the decay constant and the remaining amount of the substance, you can calculate how much time has passed. Similarly, in finance, this type of equation can model the depreciation of an asset or the decay of an investment's value over time.
To solve for t in the equation e − 0.65 t = 0.61 , take the natural logarithm of both sides, simplify it to − 0.65 t = ln ( 0.61 ) , and isolate t to find that t ≈ 0.7605 .
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