Solve for $t$. $e^{-0.24 t}=0.59$. The solution is $t = \square$. (Type an integer or decimal rounded to four decimal places as needed. Use a comma to separate answers as needed)
Answer (2)
To solve e − 0.24 t = 0.59 , take the natural logarithm of both sides, leading to t = − 0.24 l n ( 0.59 ) , which calculates to approximately 2.1983 .
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Take the natural logarithm of both sides: ln ( e − 0.24 t ) = ln ( 0.59 ) .
Simplify using logarithm properties: − 0.24 t = ln ( 0.59 ) .
Isolate t by dividing: t = − 0.24 l n ( 0.59 ) .
Calculate the value of t : t ≈ 2.1985 .
Explanation
Understanding the Problem We are given the equation e − 0.24 t = 0.59 and we need to solve for t . The variable t is in the exponent.
Taking the Natural Logarithm To solve for t , we first take the natural logarithm of both sides of the equation: ln ( e − 0.24 t ) = ln ( 0.59 ) .
Simplifying the Equation Using the property of logarithms, we simplify the left side: − 0.24 t = ln ( 0.59 ) .
Isolating t Now, we divide both sides by − 0.24 to isolate t : t = − 0.24 ln ( 0.59 ) .
Calculating the Value Calculating the value, we get: t ≈ 2.1985 .
Examples
Exponential decay is a mathematical model that describes how a quantity decreases over time. For example, the decay of a radioactive substance, the decrease in the value of a car, or the cooling of an object can be modeled using exponential decay. The formula for exponential decay is y = a e − k t , where y is the final amount, a is the initial amount, k is the decay constant, and t is the time. Solving for t in this equation allows us to determine how long it takes for the quantity to reach a certain level. This is useful in various fields, such as determining the age of artifacts using carbon dating or predicting the lifespan of a product.
