Henry places a bottle of water inside a cooler. As the water cools, its temperature C(t) in degrees Celsius is given by the following function, where t is the number of minutes since the bottle was placed in the cooler. C(t)=3+19e^(-0.045 t) Henry wants to drink the water when it reaches a temperature of 16 degrees Celsius. How many minutes should he leave it in the cooler? Round your answer to the nearest tenth, and do not round any intermediate computations. ◻minutes

Question

Henry places a bottle of water inside a cooler. As the water cools, its temperature C(t) in degrees Celsius is given by the following function, where t is the number of minutes since the bottle was placed in the cooler.  C(t)=3+19e^(-0.045 t) Henry wants to drink the water when it reaches a temperature of 16 degrees Celsius. How many minutes should he leave it in the cooler? Round your answer to the nearest tenth, and do not round any intermediate computations. ◻minutes

Bytelearn · Accepted Answer

Set Temperature Function: Set the temperature function C(t) equal to 16 degrees Celsius to solve for t. C(t) = 3 + 19e^(-0.045t) = 16Subtract to Isolate Exponential Term: Subtract 3 from both sides to isolate the exponential term. 19e^(-0.045t) = 13Divide to Solve Exponential Part: Divide both sides by 19 to solve for the exponential part. e^(-0.045t) = 13/19Take Natural Logarithm: Take the natural logarithm (ln) of both sides to solve for the exponent. ln(e^(-0.045t)) = ln(13/19)Simplify Left Side: Use the property of logarithms that ln(e^x) = x to simplify the left side. -0.045t = ln(13/19)Divide to Solve for t: Divide both sides by -0.045 to solve for t. t = ln(13/19) / -0.045Calculate Value of t: Calculate the value of t using a calculator. t ≈ ln(13/19) / -0.045 ≈ 20.1

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