A large brine tank containing a solution of salt and water is being diluted with fresh water. The relationship between the elapsed time, t, in hours, after the dilution begins, and the concentration of salt in the tank, S(t), in grams per liter (g//l), is modeled by the following function. S(t)=600*e^(-0.3 t) How many hours will it take for the concentration of salt to decrease to 100g//l ? Round your answer, if necessary, to the nearest hundredth. hours

Question

A large brine tank containing a solution of salt and water is being diluted with fresh water. The relationship between the elapsed time,  t, in hours, after the dilution begins, and the concentration of salt in the tank,  S(t), in grams per liter  (g//l), is modeled by the following function.  S(t)=600*e^(-0.3 t) How many hours will it take for the concentration of salt to decrease to  100g//l ? Round your answer, if necessary, to the nearest hundredth. hours

Bytelearn · Accepted Answer

Write Function and Concentration: Write down the given function and the concentration level we want to find the time for. We have the function S(t) = 600 * e^(-0.3t) and we want to find the time t when S(t) = 100 g/l.Set Up Equation: Set up the equation to solve for t when S(t) = 100. 100 = 600 * e^(-0.3t)Isolate Exponential Term: Divide both sides of the equation by 600 to isolate the exponential term. 100 / 600 = e^(-0.3t) 1/6 = e^(-0.3t)Take Natural Logarithm: Take the natural logarithm of both sides to solve for t. ln(1/6) = ln(e^(-0.3t))Simplify Right Side: Use the property of logarithms that ln(e^x) = x to simplify the right side of the equation. ln(1/6) = -0.3tDivide by -0.3: Divide both sides by -0.3 to solve for t. t = ln(1/6) / -0.3Calculate Value of t: Calculate the value of t using a calculator. t ≈ ln(1/6) / -0.3 ≈ ln(0.1666666667) / -0.3 ≈ 3.52636052Round to Nearest Hundredth: Round the answer to the nearest hundredth. t ≈ 3.53 hours

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