To solve the problem, we start with the information given:
1. The initial temperature of the water, [tex]\( T_0 \)[/tex], is [tex]\( 211^{\circ} F \)[/tex].
2. The ambient temperature, [tex]\( T_a \)[/tex], is [tex]\( 73^{\circ} F \)[/tex].
3. After 1.5 minutes, the temperature of the water, [tex]\( T \)[/tex], is [tex]\( 189^{\circ} F \)[/tex].
We need to determine the decay constant [tex]\( k \)[/tex] and then use this value to find the water's temperature after 4 minutes.
### Step 1: Determine [tex]\( k \)[/tex]
Using the formula from Newton's Law of Cooling:
[tex]\[ T = T_a + (T_0 - T_a) e^{-k t} \][/tex]
Substitute the known values when [tex]\( t = 1.5 \)[/tex] minutes:
[tex]\[ 189 = 73 + (211 - 73) e^{-1.5k} \][/tex]
Rearrange the equation to solve for [tex]\( k \)[/tex]:
[tex]\[ 189 - 73 = 138 e^{-1.5k} \][/tex]
[tex]\[ 116 = 138 e^{-1.5k} \][/tex]
Divide both sides by 138:
[tex]\[ \frac{116}{138} = e^{-1.5k} \][/tex]
Take the natural logarithm of both sides:
[tex]\[ \ln\left(\frac{116}{138}\right) = -1.5k \][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[ k = -\frac{\ln\left(\frac{116}{138}\right)}{1.5} \][/tex]
This calculation gives us [tex]\( k \approx 0.116 \)[/tex] (rounded to the nearest thousandth).
### Step 2: Determine the Temperature after 4 Minutes
Now that we have [tex]\( k \)[/tex], use it to find the temperature after 4 minutes.
### Substitute [tex]\( t = 4 \)[/tex] and [tex]\( k = 0.116 \)[/tex] into the formula:
[tex]\[ T = 73 + (211 - 73) e^{-0.116 \cdot 4} \][/tex]
Simplify the expression inside the exponent and solve:
[tex]\[ T = 73 + 138 e^{-0.464} \][/tex]
Calculate the exponent and complete the formula:
[tex]\[ T \approx 73 + 138 \cdot e^{-0.464} \][/tex]
Using the appropriate value for the exponential term:
[tex]\[ T \approx 73 + 138 \cdot 0.629 \][/tex]
[tex]\[ T \approx 73 + 86.802 \][/tex]
[tex]\[ T \approx 159.802 \][/tex]
Rounding to the nearest degree, the temperature of the water after 4 minutes is:
[tex]\[ 160^{\circ} F \][/tex]
Thus, the final temperature is:
[tex]\[ 160 \][/tex]
