Question

You are taking a road trip in a car without A/C. The temperture in the car is 105 degrees F. You buy a cold pop at a gas station. Its initial temperature is 45 degrees F. The pop's temperature reaches 60 degrees F after 42 minutes. Given that T 0

​
−A
T−A
​
=e −kt
where T= the temperature of the pop at time t. T 0
​
= the initial temperature of the pop. A= the temperature in the car. k= a constant that corresponds to the warming rate. and t= the length of time that the pop has been warming up. How long will it take the pop to reach a temperature of 79.75 degrees F ? It will take minutes.

Answer 1

To determine the time taken for a soda's temperature to increase from 45 degrees Fahrenheit to 79.75 degrees in a car at 105 degrees ambient temperature, one must first find the constant warming rate using the formula T = T0 + (A - T0)(1 - e^-kt), with known time and temperature values, and then use this rate to solve for the time.

Step-by-step explanation:

The student has asked about the time it will take for a cold soda pop's temperature to rise to 79.75 degrees Fahrenheit in a car without A/C, where the ambient temperature is 105 degrees Fahrenheit. We have been given that the pop's temperature reaches 60 degrees after 42 minutes, starting from an initial temperature of 45 degrees. Using the provided exponential temperature increase equation, T = T0 + (A - T0)(1 - e-kt), where T is the temperature at time t, T0 is the initial temperature, A is the ambient temperature, and k is the warming rate constant, we can solve for k and then use this value to calculate how long it will take for the pop to warm up to 79.75 degrees Fahrenheit.

First, we will find the constant k using 60 degrees as T, 45 degrees as T0, 105 degrees as A, and 42 minutes for t. Once k is found, we will use it in the equation to solve for the time t when the temperature T reaches 79.75 degrees Fahrenheit.