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A pot of piping hot stew has been removed from the stove and left to cool. The relationship between the elapsed time, m, in minutes, since the stew was removed from the stove, and the temperature of the stew, T(m), measured in ^(@)C, is modeled by the following function. T(m)=20+50*10^(-0.04 m) How many minutes will it take for the stew to cool to a temperature of 30^(@)C ? Round your answer, if necessary, to the nearest hundredth. minutes

A pot of piping hot stew has been removed from the stove and left to cool.\newlineThe relationship between the elapsed time, m m , in minutes, since the stew was removed from the stove, and the temperature of the stew, T(m) T(m) , measured in ∘C { }^{\circ} C , is modeled by the following function.\newlineT(m)=20+50⋅10−0.04m T(m)=20+50 \cdot 10^{-0.04 m} \newlineHow many minutes will it take for the stew to cool to a temperature of 30∘C 30^{\circ} \mathrm{C} ?\newlineRound your answer, if necessary, to the nearest hundredth.\newlineminutes

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Q. A pot of piping hot stew has been removed from the stove and left to cool.\newlineThe relationship between the elapsed time, m m , in minutes, since the stew was removed from the stove, and the temperature of the stew, T(m) T(m) , measured in ∘C { }^{\circ} C , is modeled by the following function.\newlineT(m)=20+50⋅10−0.04m T(m)=20+50 \cdot 10^{-0.04 m} \newlineHow many minutes will it take for the stew to cool to a temperature of 30∘C 30^{\circ} \mathrm{C} ?\newlineRound your answer, if necessary, to the nearest hundredth.\newlineminutes
  1. Set up the equation: Understand the problem and set up the equation.\newlineWe need to find the time mm when the temperature T(m)T(m) is equal to 30 ∘C30\,^\circ\text{C}. The given function is T(m)=20+50×10(−0.04m)T(m) = 20 + 50 \times 10^{(-0.04m)}. We set this equal to 30 ∘C30\,^\circ\text{C} to solve for mm.\newline30=20+50×10(−0.04m)30 = 20 + 50 \times 10^{(-0.04m)}
  2. Isolate exponential part: Isolate the exponential part of the equation.\newlineSubtract 2020 from both sides to isolate the term with the exponent on one side.\newline30−20=50×10(−0.04m)30 - 20 = 50 \times 10^{(-0.04m)}\newline10=50×10(−0.04m)10 = 50 \times 10^{(-0.04m)}
  3. Divide to solve exponential term: Divide both sides by 5050 to solve for the exponential term.\newline1050=10(−0.04m)\frac{10}{50} = 10^{(-0.04m)}\newline0.2=10(−0.04m)0.2 = 10^{(-0.04m)}
  4. Take logarithm to solve for m: Take the logarithm of both sides to solve for m.\newlineWe use the property that log⁡(ab)=b⋅log⁡(a)\log(a^b) = b \cdot \log(a) to solve for m.\newlinelog⁡(0.2)=log⁡(10−0.04m)\log(0.2) = \log(10^{-0.04m})\newlinelog⁡(0.2)=−0.04m⋅log⁡(10)\log(0.2) = -0.04m \cdot \log(10)\newlineSince log⁡(10)\log(10) is 11, we can simplify this to:\newlinelog⁡(0.2)=−0.04m\log(0.2) = -0.04m
  5. Solve for mm: Solve for mm by dividing both sides by −0.04-0.04.
    m=log⁡(0.2)−0.04m = \frac{\log(0.2)}{-0.04}
    Now we calculate the value of mm using a calculator.
    m≈log⁡(0.2)−0.04m \approx \frac{\log(0.2)}{-0.04}
    m≈−0.69897−0.04m \approx \frac{-0.69897}{-0.04}
    m≈17.47425m \approx 17.47425
  6. Round to nearest hundredth: Round the answer to the nearest hundredth.\newlinem≈17.47m \approx 17.47

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