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Answer :
To solve the problem of finding which function models the cooling of hot tea from 181 degrees Fahrenheit to 72 degrees Fahrenheit, we need to analyze the given functions and compare their results.
The functions we are evaluating are:
1. [tex]\( f(x) = 181(0.535)^t + 72 \)[/tex]
2. [tex]\( f(x) = 100(0.835)^2 + 72 \)[/tex]
3. [tex]\( f(x) = 181(0.005)^t + 72 \)[/tex]
4. [tex]\( f(x) = 109(0.065)^t + 72 \)[/tex]
We'll look into the resulting values from each option to determine the one that models the situation accurately. Here are the results:
1. For the first function, [tex]\( f(x) = 181(0.535)^t + 72 \)[/tex], the resulting value is approximately 168.835.
2. For the second function, [tex]\( f(x) = 100(0.835)^2 + 72 \)[/tex], the resulting value is approximately 141.7225.
3. For the third function, [tex]\( f(x) = 181(0.005)^t + 72 \)[/tex], the resulting value is approximately 72.905.
4. For the fourth function, [tex]\( f(x) = 109(0.065)^t + 72 \)[/tex], the resulting value is approximately 79.085.
To determine which function correctly models the cooling process, we need to consider the expected behavior of the temperature change. The tea starts at 181 degrees, and the decay rate should show how quickly the temperature reduces towards room temperature, 72 degrees, over time.
Upon examining each value:
- The third function results in a temperature close to the room temperature, which is consistent with the cooling process over time.
Therefore, the function that best represents the cooling scenario is:
[tex]\[ f(x) = 181(0.005)^t + 72 \][/tex]
This function models the situation where the hot tea is gradually cooling down to reach room temperature.
The functions we are evaluating are:
1. [tex]\( f(x) = 181(0.535)^t + 72 \)[/tex]
2. [tex]\( f(x) = 100(0.835)^2 + 72 \)[/tex]
3. [tex]\( f(x) = 181(0.005)^t + 72 \)[/tex]
4. [tex]\( f(x) = 109(0.065)^t + 72 \)[/tex]
We'll look into the resulting values from each option to determine the one that models the situation accurately. Here are the results:
1. For the first function, [tex]\( f(x) = 181(0.535)^t + 72 \)[/tex], the resulting value is approximately 168.835.
2. For the second function, [tex]\( f(x) = 100(0.835)^2 + 72 \)[/tex], the resulting value is approximately 141.7225.
3. For the third function, [tex]\( f(x) = 181(0.005)^t + 72 \)[/tex], the resulting value is approximately 72.905.
4. For the fourth function, [tex]\( f(x) = 109(0.065)^t + 72 \)[/tex], the resulting value is approximately 79.085.
To determine which function correctly models the cooling process, we need to consider the expected behavior of the temperature change. The tea starts at 181 degrees, and the decay rate should show how quickly the temperature reduces towards room temperature, 72 degrees, over time.
Upon examining each value:
- The third function results in a temperature close to the room temperature, which is consistent with the cooling process over time.
Therefore, the function that best represents the cooling scenario is:
[tex]\[ f(x) = 181(0.005)^t + 72 \][/tex]
This function models the situation where the hot tea is gradually cooling down to reach room temperature.
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