Question

the temperature of an object in degrees fahrenheit after t minutes is represented by the equation $t(t)=68e^{- 0.0174t}+72$. to the nearest degree, what is the temperature of the object after three and a half hours?

Explanation:

Step1: Convert time to minutes

Since 1 hour = 60 minutes, 3.5 hours = 3.5×60 = 210 minutes. So \(t = 210\).

Step2: Substitute \(t\) into the formula

Substitute \(t = 210\) into \(T(t)=68e^{- 0.0174t}+72\), we get \(T(210)=68e^{-0.0174\times210}+72\).
First, calculate the exponent: \(-0.0174\times210=-3.654\).
Then, find \(e^{-3.654}\). We know that \(y = e^{-3.654}=\frac{1}{e^{3.654}}\), and \(e^{3.654}\approx38.47\), so \(e^{-3.654}\approx\frac{1}{38.47}\approx0.026\).
Next, calculate \(68\times e^{-3.654}\): \(68\times0.026 = 1.768\).
Finally, find \(T(210)\): \(T(210)=1.768 + 72=73.768\).

Step3: Round the result

Rounding 73.768 to the nearest degree gives 74.

Answer:

74