High School

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Bill puts his glass of water into the refrigerator. The water's temperature starts at [tex]76^{\circ}F[/tex] and drops [tex]2^{\circ}F[/tex] every minute until it reaches the refrigerator's temperature of [tex]36^{\circ}F[/tex].

The function [tex]t(x) = 76 - 2x[/tex] describes the temperature, [tex]t(x)[/tex], that the glass of water will be after [tex]x[/tex] minutes.

What is a reasonable domain for this function?

A. [tex]36 \leq y \leq 76[/tex]
B. [tex]36 \leq x \leq 76[/tex]
C. [tex]0 \leq x \leq 20[/tex]
D. [tex]0 \leq y \leq 20[/tex]

Answer :

Sure! Let's determine the reasonable domain for the function [tex]\( t(x) = 76 - 2x \)[/tex], which describes the temperature of the glass of water after [tex]\( x \)[/tex] minutes.

### Step-by-Step Solution

1. Initial Temperature and Cooling Rate:
- The initial temperature of the water is 76°F.
- The temperature drops 2 degrees every minute.

2. Temperature Function:
- The function provided is [tex]\( t(x) = 76 - 2x \)[/tex], where [tex]\( t(x) \)[/tex] represents the temperature of the water after [tex]\( x \)[/tex] minutes.

3. Refrigerator's Temperature:
- The refrigerator's temperature is 36°F.
- The water will keep cooling until it reaches this temperature.

4. Finding Time to Reach Refrigerator Temperature:
- We need to find when the water's temperature will be 36°F.
- Set [tex]\( t(x) = 36 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[
36 = 76 - 2x
\][/tex]
[tex]\[
36 - 76 = -2x
\][/tex]
[tex]\[
-40 = -2x
\][/tex]
[tex]\[
x = \frac{-40}{-2} = 20
\][/tex]

5. Reasonable Domain for the Function:
- [tex]\( x \)[/tex] represents the number of minutes after Bill places the glass in the refrigerator.
- The process starts at [tex]\( x = 0 \)[/tex] minutes and continues until the temperature is 36°F.
- Therefore, the function [tex]\( t(x) \)[/tex] is valid from [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex].

So, the reasonable domain for the function [tex]\( t(x) = 76 - 2x \)[/tex] is [tex]\( 0 \leq x \leq 20 \)[/tex].

Thus, the correct answer is:
[tex]\[ 0 \leq x \leq 20 \][/tex]

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