A pot of soup, currently 74°C above room temperature, is left out to cool. If that temperature difference decreases by 5% per minute, what will the difference be in 14 minutes? If... A pot of soup, currently 74°C above room temperature, is left out to cool. If that temperature difference decreases by 5% per minute, what will the difference be in 14 minutes? If necessary, round your answer to the nearest tenth.
Understand the Problem
The question is asking how much the temperature difference between a pot of soup and room temperature will decrease after 14 minutes, given that the difference decreases by 5% each minute. The main approach involves calculating the decreasing percentage iteratively over the specified time period.
Answer
The temperature difference after 14 minutes will be approximately $36.1 \text{°C}$.
Answer for screen readers
The temperature difference after 14 minutes will be approximately $36.1 \text{°C}$.
Steps to Solve
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Initial Setup Start with the initial temperature difference of the soup above room temperature. $$ T_0 = 74 \text{°C} $$
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Determine the Rate of Decrease The temperature difference decreases by 5% each minute. This means after each minute, the remaining temperature difference is 95% of the previous minute’s difference.
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Mathematical Representation You can represent the temperature difference after $n$ minutes as: $$ T_n = T_0 \times (0.95)^n $$
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Calculate the Temperature Difference After 14 Minutes Substitute $n = 14$ into the equation to find the temperature difference: $$ T_{14} = 74 \times (0.95)^{14} $$
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Perform the Calculation First, calculate $(0.95)^{14}$, then multiply by 74 to find the difference.
$$ T_{14} \approx 74 \times 0.4876 \approx 36.1 \text{°C} $$
The temperature difference after 14 minutes will be approximately $36.1 \text{°C}$.
More Information
This problem demonstrates exponential decay, where a quantity decreases by a fixed percentage over time. The method used is common in cooling processes, which can be seen in practical applications ranging from food preparation to scientific experiments.
Tips
- Misunderstanding Percentage Decrease: A common mistake is to subtract 5% from the initial value instead of multiplying by 0.95 for each minute.
- Wrong Calculation of Powers: Another mistake can arise from incorrectly calculating powers of $0.95$. Make sure to use accurate methods or a calculator.
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