Questions: A bottle of ginger ale initially has a temperature of 75°F. It is left to cool in a refrigerator that has a temperature of 36°F. After 10 minutes the temperature of the ginger ale is 62°F. Complete parts a through c. a. Use Newton's Law of Cooling, T = C + (T₀ - C)eᵏᵗ, to find a model for the temperature of the ginger ale, T, after t minutes. b. What is the temperature of the ginger ale after 15 minutes? c. When will the temperature of the ginger ale be 51°F?

Solve for the time when the temperature of the ginger ale is \(51^{\circ} \mathrm{F}\).

Set up the equation using the temperature model.

We are given the temperature model \( T = 36 + 39e^{-0.0405t} \) and we want to find the time \( t \) when the temperature \( T \) is \( 51^{\circ} \mathrm{F} \). Set up the equation:
\[ 51 = 36 + 39e^{-0.0405t} \]

Isolate the exponential term.

Subtract 36 from both sides of the equation:
\[ 15 = 39e^{-0.0405t} \]

Solve for the exponential term.

Divide both sides by 39:
\[ \frac{15}{39} = e^{-0.0405t} \]

Take the natural logarithm of both sides.

Apply the natural logarithm to both sides:
\[ \ln\left(\frac{15}{39}\right) = -0.0405t \]

Solve for \( t \).

Divide by \(-0.0405\) to solve for \( t \):
\[ t = \frac{\ln\left(\frac{15}{39}\right)}{-0.0405} \]
Using a calculator, we find:
\[ t \approx 24 \]

The temperature of the ginger ale will be \( 51^{\circ} \mathrm{F} \) after approximately \(\boxed{24}\) minutes.

The temperature of the ginger ale will be \( 51^{\circ} \mathrm{F} \) after approximately \(\boxed{24}\) minutes.