Question

What is the meaning of the statement ((30) =76?

Answer 1

A) 30 minutes after the air conditioner started to cool the house, the temperature inside the house has dropped to 76°F

B)
`f(x)=-0.4x+88`

Explanation:

A)

In this problem, f(x) represents temperature, in degrees Fahrenheit, of Sam’s house x minutes after the air conditioner started to cool the house.

So we have:

x = minutes after the air conditioner started to cool the house

f(x) = temperature

Here we want to find what is the meaning of

`f(30)=76`

Which implies that:

`x=30`, which means that 30 minutes have passed after the conditioner has started to cool the house

`f(x)=76`, which means that the temperature of the room at that time is 76°F

So, the interpretation of f(30) =76 is that:

"30 minutes after the air conditioner started to cool the house, the temperature inside the house has dropped to 76°F"

B)

Here we want to write explicitly a function that tells us how the temperature inside the house varies as a function of x, the number of minutes passed after the air conditioner has turned on.

We know the following information:

- At time x = 0, the temperature inside the house is 88°F

- At time x = 30 min, the temperature inside the house is 76°F

We can assume that the temperature inside the house changes linearly with time; so, we can describe the temperature function with the equation of a line:

`f(x)=mx+q`

where

`q` is the value of the temperature at x = 0, so q = 88

m is the rate of change of the temperature

The rate of change can be found by calculating the rate of change of the temperature with respect to the change in time; so we find:

`m=(76-88)/(30-0) =-0.4`

which means that the temperature decreases by 0.4°F every minute. So, the function can be written as

`f(x)=-0.4x+88`