To solve the problem, follow these steps:
1. Understand the problem:
We need to find the temperature [tex]\( T \)[/tex] of an object after 1.5 hours using the given temperature function [tex]\( T(t) = 65 e^{-0.0174 t} + 72 \)[/tex].
2. Convert hours to minutes:
Since the given formula uses time [tex]\( t \)[/tex] in minutes, convert 1.5 hours to minutes:
[tex]\[
t = 1.5 \text{ hours} \times 60 \frac{\text{minutes}}{\text{hour}} = 90 \text{ minutes}
\][/tex]
3. Substitute [tex]\( t = 90 \)[/tex] into the formula:
Now, substitute [tex]\( t = 90 \)[/tex] minutes into the equation:
[tex]\[
T(90) = 65 e^{-0.0174 \times 90} + 72
\][/tex]
4. Compute the temperature [tex]\( T(90) \)[/tex]:
Evaluating the expression gives:
[tex]\[
T(90) \approx 85.57713692850301
\][/tex]
5. Round to the nearest degree:
The nearest integer to [tex]\( 85.57713692850301 \)[/tex] is 86 degrees.
So, the temperature of the object after 1.5 hours is approximately [tex]\( 86^{\circ} F \)[/tex].
