Answer :
To solve this problem, we need to find out which system of equations represents the time it will take for the cupcakes and cake to reach the same temperature.
### Step-by-step Explanation:
1. Understanding the Problem:
- Cupcakes: When placed in the refrigerator, they have a temperature of [tex]\(121^\circ F\)[/tex] and cool down at a rate of [tex]\(21^\circ F\)[/tex] per minute.
- Cake: When placed in the freezer, it has a temperature of [tex]\(145^\circ F\)[/tex] and cools down at a rate of [tex]\(5.4^\circ F\)[/tex] per minute.
2. Setting Up the Equations:
- For the cupcakes, their temperature [tex]\(y\)[/tex] after [tex]\(x\)[/tex] minutes can be represented by the equation:
[tex]\[
y = 121 - 21x
\][/tex]
- For the cake, its temperature [tex]\(y\)[/tex] after [tex]\(x\)[/tex] minutes can be represented by the equation:
[tex]\[
y = 145 - 5.4x
\][/tex]
- Since we are asked for a system of equations that shows when the temperatures are the same, we rearrange the equations in the form:
[tex]\[
y = -21x + 121
\][/tex]
[tex]\[
y = -5.4x + 145
\][/tex]
3. Choosing the Correct System of Equations:
- The given options are:
- [tex]\(\left\{\begin{array}{l}y=-2.1x+121 \\ y=-5.4x+145\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}y=2.1x+121 \\ y=5.4x+145\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}y=-2.1x+145 \\ y=-5.4x+121\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}y=2.1x+145 \\ y=5.4x+121\end{array}\right.\)[/tex]
- Given the cooling rates and initial temperatures, the correct system of equations is:
- [tex]\(\left\{\begin{array}{l}y=-21x+121 \\ y=-5.4x+145\end{array}\right.\)[/tex]
4. Matching the Correct System:
- However, noticing there is an error in the given configurations. The correct expressions should include [tex]\(y = -21x + 121\)[/tex] and [tex]\(y = -5.4x + 145\)[/tex], based on actual cooling rates in the context of the problem.
- Despite this, based on the expected configuration and completed Python output being noticed, the setup selected wrongly was [tex]\(\left\{\begin{array}{l}y=-2.1x+121 \\ y=-5.4x+145\end{array}\right.\)[/tex], albeit with correct logical derivation and conclusion requiring standard cooling rate figures.
### Final Result:
The system of equations representing the time [tex]\(x\)[/tex] it will take for the cupcakes and the cake to be the same temperature is:
[tex]\[
\left\{\begin{array}{l}y=-21x+121 \\ y=-5.4x+145\end{array}\right.
\][/tex]
### Step-by-step Explanation:
1. Understanding the Problem:
- Cupcakes: When placed in the refrigerator, they have a temperature of [tex]\(121^\circ F\)[/tex] and cool down at a rate of [tex]\(21^\circ F\)[/tex] per minute.
- Cake: When placed in the freezer, it has a temperature of [tex]\(145^\circ F\)[/tex] and cools down at a rate of [tex]\(5.4^\circ F\)[/tex] per minute.
2. Setting Up the Equations:
- For the cupcakes, their temperature [tex]\(y\)[/tex] after [tex]\(x\)[/tex] minutes can be represented by the equation:
[tex]\[
y = 121 - 21x
\][/tex]
- For the cake, its temperature [tex]\(y\)[/tex] after [tex]\(x\)[/tex] minutes can be represented by the equation:
[tex]\[
y = 145 - 5.4x
\][/tex]
- Since we are asked for a system of equations that shows when the temperatures are the same, we rearrange the equations in the form:
[tex]\[
y = -21x + 121
\][/tex]
[tex]\[
y = -5.4x + 145
\][/tex]
3. Choosing the Correct System of Equations:
- The given options are:
- [tex]\(\left\{\begin{array}{l}y=-2.1x+121 \\ y=-5.4x+145\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}y=2.1x+121 \\ y=5.4x+145\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}y=-2.1x+145 \\ y=-5.4x+121\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}y=2.1x+145 \\ y=5.4x+121\end{array}\right.\)[/tex]
- Given the cooling rates and initial temperatures, the correct system of equations is:
- [tex]\(\left\{\begin{array}{l}y=-21x+121 \\ y=-5.4x+145\end{array}\right.\)[/tex]
4. Matching the Correct System:
- However, noticing there is an error in the given configurations. The correct expressions should include [tex]\(y = -21x + 121\)[/tex] and [tex]\(y = -5.4x + 145\)[/tex], based on actual cooling rates in the context of the problem.
- Despite this, based on the expected configuration and completed Python output being noticed, the setup selected wrongly was [tex]\(\left\{\begin{array}{l}y=-2.1x+121 \\ y=-5.4x+145\end{array}\right.\)[/tex], albeit with correct logical derivation and conclusion requiring standard cooling rate figures.
### Final Result:
The system of equations representing the time [tex]\(x\)[/tex] it will take for the cupcakes and the cake to be the same temperature is:
[tex]\[
\left\{\begin{array}{l}y=-21x+121 \\ y=-5.4x+145\end{array}\right.
\][/tex]
