A mug of warm apple cider is gradually cooling. Its temperature in degrees Celsius (^(@)C:} ) can be modeled with the expression 23+24*2^(-0.014 t), where the variable t represents the time in minutes. Approximately how many minutes will it take for the temperature to reach 29^(@)C ? (Round to the nearest minute.)

Question

A mug of warm apple cider is gradually cooling. Its temperature in degrees Celsius  (^(@)C:} ) can be modeled with the expression  23+24*2^(-0.014 t), where the variable  t represents the time in minutes. Approximately how many minutes will it take for the temperature to reach  29^(@)C ? (Round to the nearest minute.)

Bytelearn · Accepted Answer

Write Equation: Write down the given temperature model equation. We have the temperature model equation: T(t) = 23 + 24 * 2^(-0.014t), where T(t) is the temperature in degrees Celsius at time t minutes.Set Equal to 29: Set the temperature model equation equal to 29 degrees Celsius to find the time t when the temperature will be 29 degrees Celsius. 29 = 23 + 24 * 2^(-0.014t)Subtract and Isolate: Subtract 23 from both sides of the equation to isolate the exponential term. 29 - 23 = 24 * 2^(-0.014t) 6 = 24 * 2^(-0.014t)Divide and Solve: Divide both sides of the equation by 24 to solve for the exponential term. 6 / 24 = 2^(-0.014t) 1/4 = 2^(-0.014t)Take Logarithm: Take the logarithm of both sides of the equation to solve for t. We will use the natural logarithm (ln) and the property that ln(a^b) = b * ln(a). ln(1/4) = ln(2^(-0.014t)) ln(1/4) = -0.014t * ln(2)Divide and Calculate: Divide both sides of the equation by -0.014 * ln(2) to solve for t. t = ln(1/4) / (-0.014 * ln(2))Calculate t: Calculate the value of t using a calculator. t ≈ ln(0.25) / (-0.014 * ln(2)) t ≈ -1.38629436112 / (-0.014 * 0.69314718056) t ≈ -1.38629436112 / (-0.00970303639488) t ≈ 142.857142857Round to Nearest Minute: Round the value of t to the nearest minute. t ≈ 143 minutes

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