Answer :
Certainly! Let's solve the question step-by-step to find the temperature of the coffee at [tex]\( t = 1 \)[/tex] minute.
1. Understand the Problem:
- We start with a cup of coffee at an initial temperature of 120 degrees Fahrenheit.
- The rate at which the temperature decreases is given by the function [tex]\( r(t) = 55 e^{-0.03 t^2} \)[/tex], where [tex]\( t \)[/tex] is the time in minutes.
2. Find the decrease in temperature at [tex]\( t = 1 \)[/tex] minute:
- We want to determine the change in temperature from [tex]\( t = 0 \)[/tex] to [tex]\( t = 1 \)[/tex] by evaluating the rate at [tex]\( t = 1 \)[/tex].
- Plug [tex]\( t = 1 \)[/tex] into the rate function:
[tex]\[
r(1) = 55 e^{-0.03 \times 1^2}
\][/tex]
- Calculating the rate:
[tex]\[
r(1) = 55 e^{-0.03} \approx 53.37
\][/tex]
3. Calculate the temperature at [tex]\( t = 1 \)[/tex] minute:
- The initial temperature is 120 degrees Fahrenheit.
- The temperature decreases by approximately 53.37 degrees after 1 minute.
- So, the temperature at [tex]\( t = 1 \)[/tex] minute is:
[tex]\[
\text{Temperature at } t = 1 = 120 - 53.37 \approx 66.63
\][/tex]
4. Round the final answer:
- The closest option to this calculated temperature is (D) 66.6 degrees Fahrenheit.
Therefore, the temperature of the coffee at [tex]\( t = 1 \)[/tex] minute is approximately [tex]\( 66.6^{\circ} F \)[/tex].
1. Understand the Problem:
- We start with a cup of coffee at an initial temperature of 120 degrees Fahrenheit.
- The rate at which the temperature decreases is given by the function [tex]\( r(t) = 55 e^{-0.03 t^2} \)[/tex], where [tex]\( t \)[/tex] is the time in minutes.
2. Find the decrease in temperature at [tex]\( t = 1 \)[/tex] minute:
- We want to determine the change in temperature from [tex]\( t = 0 \)[/tex] to [tex]\( t = 1 \)[/tex] by evaluating the rate at [tex]\( t = 1 \)[/tex].
- Plug [tex]\( t = 1 \)[/tex] into the rate function:
[tex]\[
r(1) = 55 e^{-0.03 \times 1^2}
\][/tex]
- Calculating the rate:
[tex]\[
r(1) = 55 e^{-0.03} \approx 53.37
\][/tex]
3. Calculate the temperature at [tex]\( t = 1 \)[/tex] minute:
- The initial temperature is 120 degrees Fahrenheit.
- The temperature decreases by approximately 53.37 degrees after 1 minute.
- So, the temperature at [tex]\( t = 1 \)[/tex] minute is:
[tex]\[
\text{Temperature at } t = 1 = 120 - 53.37 \approx 66.63
\][/tex]
4. Round the final answer:
- The closest option to this calculated temperature is (D) 66.6 degrees Fahrenheit.
Therefore, the temperature of the coffee at [tex]\( t = 1 \)[/tex] minute is approximately [tex]\( 66.6^{\circ} F \)[/tex].
