Question

While Sam was at work, his house lost electrical power. By the time the electrical power came back on, the temperature inside the house was
88°F. The air conditioner immediately started to cool the house.

Let f(x) represent the temperature, in degrees Fahrenheit, of Sam’s house x minutes after the air conditioner started to cool the house.

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

B. Use function notation to represent the temperature of the house when the air conditioner started to cool the house.

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

Step-by-step 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 explicitely 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 hosue 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:

$m=\frac{76-88}{30-0}=-0.4$

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

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