Question

The school assembly is being held over the lunch hour in the school gym. All the teachers and students are there by noon and the assembly begins. About 45 minutes after the assembly begins, the temperature within the gym remains a steady 77 degrees Fahrenheit for a few minutes. As the students leave after the assembly ends at the end of the hour, the gym begins to slowly cool down.

Let T denote the temperature of the gym in degrees Fahrenheit and M denote the time, in minutes, since noon.

A. Is M a function of T? Explain why or why not
B. Explain why T is a function of M, and consider the function T=g(M)
C. Interpret the meaning of g (0) in the context of the problem.
D. Becky says: “The temperature increased 5 degrees in the first half hour after the assembly began.” Which of the following equations best represents this statement? Explain your choice.
1. g(30)=5
2.
`(g(30)-g(0))/(30)`=5
3. g(30)-g(0)=5
4. T=g(30)-5

Answer 1

Answer: A. Is M a function of T? Explain why or why not.

No, M is not a function of T. A function is a rule that assigns a unique output (in this case, time M) to each input (in this case, temperature T). However, in the given scenario, different times M can have the same temperature T. For example, after 45 minutes, the temperature T is 77 degrees Fahrenheit, but there are two different times (45 minutes and 75 minutes) that have the same temperature. Therefore, M is not a function of T.

B. Explain why T is a function of M, and consider the function T=g(M).

Yes, T is a function of M. In the context of this problem, as time (M) increases, the temperature (T) changes accordingly. Each time M corresponds to a unique temperature T, and there is only one temperature at any given time. Therefore, T can be represented as a function g(M) where the temperature (T) is the output, and time (M) is the input.

C. Interpret the meaning of g(0) in the context of the problem.

In the context of the problem, g(0) represents the temperature of the gym (T) at time 0, which is the time of noon when the school assembly begins. It means that at the start of the assembly (noon), the temperature T in the gym is given by g(0). This value may be equal to the temperature at noon or the initial temperature before the assembly begins.

D. Becky says: “The temperature increased 5 degrees in the first half hour after the assembly began.” Which of the following equations best represents this statement? Explain your choice.

The correct equation that represents Becky's statement is:

g(30) - g(0) = 5

Step-by-step explanation:

g(30) represents the temperature at 30 minutes after the assembly began, and g(0) represents the temperature at the start of the assembly (noon). So, g(30) - g(0) represents the change in temperature between these two times. According to Becky's statement, this change in temperature is 5 degrees, which matches the right-hand side of the equation.