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

[tex]t\:\:\:\:\:\:\:40\:\:\:\:\:\:\:\:\:\:50\:\:\:\:\:\:\:\:\:\:60\:\:\:\:\:\:\:\:\:\:70\:\:\:\:\:\:\:\:\:\:\:80\:\:\:\:\:\:\:\:\:\:90\:\:\:\:\:\:\:\:\:\:\:100[/tex]

[tex]c\:\:\:\:\:\:\:\:10\:\:\:\:\:\:\:\:50\:\:\:\:\:\:\:\:\:\:\:90\:\:\:\:\:\:\:\:\:130\:\:\:\:\:\:\:\:\:170\:\:\:\:\:\:\:\:\:210\:\:\:\:\:\:\:\:\:\:250[/tex]

PART 1)

Considering the function

[tex]c\:=\:4t\:-\:150[/tex]

As we know that

when [tex]t = 60[/tex]

then [tex]c = 4(60) - 150 = 240 - 150 = 90[/tex]

Meaning the number of chirps per minute would increase to [tex]90[/tex], 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

[tex](40, 10)[/tex]

[tex](50, 50)[/tex]

[tex]\mathrm{Slope\:between\:two\:points}:\mathrm{Slope}=\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]\left(x_1,\:y_1\right)=\left(40,\:10\right),\:\left(x_2,\:y_2\right)=\left(50,\:50\right)[/tex]

[tex]m=\frac{50-10}{50-40}[/tex]

[tex]m=4[/tex]

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

Thus, the rate of change will be [tex]4[/tex].

Part 3)

Considering the function

[tex]c\:=\:4t\:-\:150[/tex]

The data table for this function

[tex]t\:\:\:\:\:\:\:40\:\:\:\:\:\:\:\:\:\:50\:\:\:\:\:\:\:\:\:\:60\:\:\:\:\:\:\:\:\:\:70\:\:\:\:\:\:\:\:\:\:\:80\:\:\:\:\:\:\:\:\:\:90\:\:\:\:\:\:\:\:\:\:\:100[/tex]

[tex]c\:\:\:\:\:\:\:\:10\:\:\:\:\:\:\:\:50\:\:\:\:\:\:\:\:\:\:\:90\:\:\:\:\:\:\:\:\:130\:\:\:\:\:\:\:\:\:170\:\:\:\:\:\:\:\:\:210\:\:\:\:\:\:\:\:\:\:250[/tex]

Putting [tex]t = 40[/tex] in the function brings the value of [tex]c[/tex] as [tex]10[/tex].

i.e.

[tex]c\:=\:4\left(40\right)\:-\:150=160=10[/tex]

Yes, it does make sense.

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