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Answer :
To solve this problem, we need to interpret the given statements about the function [tex]\( f(t) \)[/tex], which represents the temperature in degrees Fahrenheit at [tex]\( t \)[/tex] hours after 6 a.m.
Let's look at each statement:
A. [tex]\( f(0)=76 \)[/tex]; The temperature at 6:00 am is [tex]\( 76^{\circ} \text{F} \)[/tex]
- Since [tex]\( f(t) \)[/tex] gives the temperature [tex]\( t \)[/tex] hours after 6 a.m., [tex]\( f(0) \)[/tex] represents the temperature at 6 a.m.
- Therefore, this statement is correct. The temperature at 6:00 a.m. is [tex]\( 76^{\circ} \text{F} \)[/tex].
B. [tex]\( f(t)=84^{\circ} \text{F} \)[/tex]; After [tex]\( t \)[/tex] hours the temperature will be [tex]\( 84^{\circ} \text{F} \)[/tex]
- This statement indicates that for some value of [tex]\( t \)[/tex], [tex]\( f(t) \)[/tex] equals [tex]\( 84^{\circ} \text{F} \)[/tex].
- Without knowing the specific value of [tex]\( t \)[/tex], it simply states that at some point, the temperature is [tex]\( 84^{\circ} \text{F} \)[/tex]. This does not provide enough specific information to confirm it as a unique interpretation, so it is not selected.
C. [tex]\( f(3)=79 \)[/tex]; The temperature at 3:00 pm will be [tex]\( 79^{\circ} \text{F} \)[/tex]
- [tex]\( f(3) \)[/tex] refers to the temperature 3 hours after 6 a.m., which is 9 a.m., not 3 p.m.
- Therefore, this statement is incorrect.
D. [tex]\( f(3)=79 \)[/tex]; The temperature at 9:00 am will be [tex]\( 79^{\circ} \text{F} \)[/tex]
- As previously mentioned, [tex]\( f(3) \)[/tex] represents the temperature 3 hours after 6 a.m., which is indeed 9 a.m.
- Therefore, this statement is correct. The temperature at 9:00 a.m. is [tex]\( 79^{\circ} \text{F} \)[/tex].
E. [tex]\( f(10)>f(12) \)[/tex]; The temperature at 10:00 am is greater than the temperature at 12:00 pm
- This statement is comparative. It suggests that the temperature at 10 a.m. is higher than the temperature at 12 p.m.
- Since it involves a comparison and doesn't conflict with any logical assumptions about temperatures varying throughout the day, it can be considered correct.
Based on this analysis, the correct interpretations are statements A, D, and E.
Let's look at each statement:
A. [tex]\( f(0)=76 \)[/tex]; The temperature at 6:00 am is [tex]\( 76^{\circ} \text{F} \)[/tex]
- Since [tex]\( f(t) \)[/tex] gives the temperature [tex]\( t \)[/tex] hours after 6 a.m., [tex]\( f(0) \)[/tex] represents the temperature at 6 a.m.
- Therefore, this statement is correct. The temperature at 6:00 a.m. is [tex]\( 76^{\circ} \text{F} \)[/tex].
B. [tex]\( f(t)=84^{\circ} \text{F} \)[/tex]; After [tex]\( t \)[/tex] hours the temperature will be [tex]\( 84^{\circ} \text{F} \)[/tex]
- This statement indicates that for some value of [tex]\( t \)[/tex], [tex]\( f(t) \)[/tex] equals [tex]\( 84^{\circ} \text{F} \)[/tex].
- Without knowing the specific value of [tex]\( t \)[/tex], it simply states that at some point, the temperature is [tex]\( 84^{\circ} \text{F} \)[/tex]. This does not provide enough specific information to confirm it as a unique interpretation, so it is not selected.
C. [tex]\( f(3)=79 \)[/tex]; The temperature at 3:00 pm will be [tex]\( 79^{\circ} \text{F} \)[/tex]
- [tex]\( f(3) \)[/tex] refers to the temperature 3 hours after 6 a.m., which is 9 a.m., not 3 p.m.
- Therefore, this statement is incorrect.
D. [tex]\( f(3)=79 \)[/tex]; The temperature at 9:00 am will be [tex]\( 79^{\circ} \text{F} \)[/tex]
- As previously mentioned, [tex]\( f(3) \)[/tex] represents the temperature 3 hours after 6 a.m., which is indeed 9 a.m.
- Therefore, this statement is correct. The temperature at 9:00 a.m. is [tex]\( 79^{\circ} \text{F} \)[/tex].
E. [tex]\( f(10)>f(12) \)[/tex]; The temperature at 10:00 am is greater than the temperature at 12:00 pm
- This statement is comparative. It suggests that the temperature at 10 a.m. is higher than the temperature at 12 p.m.
- Since it involves a comparison and doesn't conflict with any logical assumptions about temperatures varying throughout the day, it can be considered correct.
Based on this analysis, the correct interpretations are statements A, D, and E.
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