Question

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

Answer 1

Step-by-step 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

• $(40, 10)$

• $(50, 50)$

$\mathrm{Slope\:between\:two\:points}:\mathrm{Slope}=\frac{y_2-y_1}{x_2-x_1}$

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

$m=\frac{50-10}{50-40}$

$m=4$

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 $10$.