Question

What is the rate of change for this function / careful / how can it be interpreted using the problem context?

Answer 1

Explanation:

FORMATION OF TABLE FOR THE FUNCTION
`c\:=\:4t\:-\:150`

As t represents the temperature in degrees Fahrenheit and c represents the number of cricket chirps per minute.

Considering the function

`c\:=\:4t\:-\:150`

when
`t = 40`

then
`c = 4(40) - 150 = 160 - 150 = 10`

when
`t = 50`

then
`c = 4(50) - 150 = 200 - 150 = 50`

when
`t = 60`

then
`c = 4(60) - 150 = 240 - 150 = 90`

when
`t = 70`

then
`c = 4(70) - 150 = 280 - 150 = 130`

when
`t = 80`

then
`c = 4(80) - 150 = 320 - 150 = 170`

when
`t = 90`

then
`c = 4(90) - 150 = 360 - 150 = 210`

when
`t = 100`

then
`c = 4(100) - 150 = 400 - 150 = 250`

So

Lets form the data table for this function based on the determined values

`t\:\:\:\:\:\:\:40\:\:\:\:\:\:\:\:\:\:50\:\:\:\:\:\:\:\:\:\:60\:\:\:\:\:\:\:\:\:\:70\:\:\:\:\:\:\:\:\:\:\:80\:\:\:\:\:\:\:\:\:\:90\:\:\:\:\:\:\:\:\:\:\:100`

`c\:\:\:\:\:\:\:\:10\:\:\:\:\:\:\:\:50\:\:\:\:\:\:\:\:\:\:\:90\:\:\:\:\:\:\:\:\:130\:\:\:\:\:\:\:\:\:170\:\:\:\:\:\:\:\:\:210\:\:\:\:\:\:\:\:\:\:250`

PART 1)

Considering the function

`c\:=\:4t\:-\:150`

As we know that

when
`t = 60`

then
`c = 4(60) - 150 = 240 - 150 = 90`

• Meaning the number of chirps per minute would increase to
  `90`, when the temperature t in degrees Fahrenheit increase to 60.

The appropriate logic is that the speed at which cricket chirps is based on the temperature. The table table also indicates that as the temperature t increases, the number of cricket chirps also increases.

PART 2)

• A rate of change is a rate that determines how one quantity changes in relation to another quantity.

Considering the two points

• 
  `(40, 10)`

• 
  `(50, 50)`

`\mathrm{Slope\:between\:two\:points}:\mathrm{Slope}=(y_2-y_1)/(x_2-x_1)`

`\left(x_1,\:y_1\right)=\left(40,\:10\right),\:\left(x_2,\:y_2\right)=\left(50,\:50\right)`

`m=(50-10)/(50-40)`

`m=4`

It logically means for every increase of
`10` units in
`t` (temperature in degrees Fahrenheit), the value of
`c` (number of chirps) is increasing to
`40` units.

Thus, the rate of change will be
`4`.

Part 3)

Considering the function

`c\:=\:4t\:-\:150`

The data table for this function

`t\:\:\:\:\:\:\:40\:\:\:\:\:\:\:\:\:\:50\:\:\:\:\:\:\:\:\:\:60\:\:\:\:\:\:\:\:\:\:70\:\:\:\:\:\:\:\:\:\:\:80\:\:\:\:\:\:\:\:\:\:90\:\:\:\:\:\:\:\:\:\:\:100`

`c\:\:\:\:\:\:\:\:10\:\:\:\:\:\:\:\:50\:\:\:\:\:\:\:\:\:\:\:90\:\:\:\:\:\:\:\:\:130\:\:\:\:\:\:\:\:\:170\:\:\:\:\:\:\:\:\:210\:\:\:\:\:\:\:\:\:\:250`

Putting
`t = 40` in the function brings the value of
`c` as
`10`.

i.e.

`c\:=\:4\left(40\right)\:-\:150=160=10`

Yes, it does make sense.

Its logical meaning is that at the start, when the value of
`t` was
`40` temperature in degrees Fahrenheit, then the value of
`c` (number of chirps per minute) was
`10`.